b+tree.ts

// B+ tree by David Piepgrass. License: MIT
import { ISortedMap, ISortedMapF, ISortedSet } from './interfaces';

export {
  ISetSource, ISetSink, ISet, ISetF, ISortedSetSource, ISortedSet, ISortedSetF,
  IMapSource, IMapSink, IMap, IMapF, ISortedMapSource, ISortedMap, ISortedMapF
} from './interfaces';

export type EditRangeResult<V,R=number> = {value?:V, break?:R, delete?:boolean};

type index = number;

// Informative microbenchmarks & stuff:
// http://www.jayconrod.com/posts/52/a-tour-of-v8-object-representation (very educational)
// https://blog.mozilla.org/luke/2012/10/02/optimizing-javascript-variable-access/ (local vars are faster than properties)
// http://benediktmeurer.de/2017/12/13/an-introduction-to-speculative-optimization-in-v8/ (other stuff)
// https://jsperf.com/js-in-operator-vs-alternatives (avoid 'in' operator; `.p!==undefined` faster than `hasOwnProperty('p')` in all browsers)
// https://jsperf.com/instanceof-vs-typeof-vs-constructor-vs-member (speed of type tests varies wildly across browsers)
// https://jsperf.com/detecting-arrays-new (a.constructor===Array is best across browsers, assuming a is an object)
// https://jsperf.com/shallow-cloning-methods (a constructor is faster than Object.create; hand-written clone faster than Object.assign)
// https://jsperf.com/ways-to-fill-an-array (slice-and-replace is fastest)
// https://jsperf.com/math-min-max-vs-ternary-vs-if (Math.min/max is slow on Edge)
// https://jsperf.com/array-vs-property-access-speed (v.x/v.y is faster than a[0]/a[1] in major browsers IF hidden class is constant)
// https://jsperf.com/detect-not-null-or-undefined (`x==null` slightly slower than `x===null||x===undefined` on all browsers)
// Overall, microbenchmarks suggest Firefox is the fastest browser for JavaScript and Edge is the slowest.
// Lessons from https://v8project.blogspot.com/2017/09/elements-kinds-in-v8.html:
//   - Avoid holes in arrays. Avoid `new Array(N)`, it will be "holey" permanently.
//   - Don't read outside bounds of an array (it scans prototype chain).
//   - Small integer arrays are stored differently from doubles
//   - Adding non-numbers to an array deoptimizes it permanently into a general array
//   - Objects can be used like arrays (e.g. have length property) but are slower
//   - V8 source (NewElementsCapacity in src/objects.h): arrays grow by 50% + 16 elements

/**
 * Types that BTree supports by default
 */
export type DefaultComparable = number | string | Date | boolean | null | undefined | (number | string)[] |
               { valueOf: () => number | string | Date | boolean | null | undefined | (number | string)[] };

/**
 * Compares DefaultComparables to form a strict partial ordering.
 * 
 * Handles +/-0 and NaN like Map: NaN is equal to NaN, and -0 is equal to +0.
 * 
 * Arrays are compared using '<' and '>', which may cause unexpected equality:
 * for example [1] will be considered equal to ['1'].
 * 
 * Two objects with equal valueOf compare the same, but compare unequal to
 * primitives that have the same value.
 */
export function defaultComparator(a: DefaultComparable, b: DefaultComparable): number {
  // Special case finite numbers first for performance.
  // Note that the trick of using 'a - b' and checking for NaN to detect non-numbers
  // does not work if the strings are numeric (ex: "5"). This would leading most 
  // comparison functions using that approach to fail to have transitivity.
  if (Number.isFinite(a as any) && Number.isFinite(b as any)) {
    return a as number - (b as number);
  }

  // The default < and > operators are not totally ordered. To allow types to be mixed
  // in a single collection, compare types and order values of different types by type.
  let ta = typeof a;
  let tb = typeof b;
  if (ta !== tb) {
    return ta < tb ? -1 : 1;
  }

  if (ta === 'object') {
    // standardized JavaScript bug: null is not an object, but typeof says it is
    if (a === null)
      return b === null ? 0 : -1;
    else if (b === null)
      return 1;

    a = a!.valueOf() as DefaultComparable;
    b = b!.valueOf() as DefaultComparable;
    ta = typeof a;
    tb = typeof b;
    // Deal with the two valueOf()s producing different types
    if (ta !== tb) {
      return ta < tb ? -1 : 1;
    }
  }

  // a and b are now the same type, and will be a number, string or array 
  // (which we assume holds numbers or strings), or something unsupported.
  if (a! < b!) return -1;
  if (a! > b!) return 1;
  if (a === b) return 0;

  // Order NaN less than other numbers
  if (Number.isNaN(a as any))
    return Number.isNaN(b as any) ? 0 : -1;
  else if (Number.isNaN(b as any))
    return 1;
  // This could be two objects (e.g. [7] and ['7']) that aren't ordered
  return Array.isArray(a) ? 0 : Number.NaN;
};

/**
 * Compares items using the < and > operators. This function is probably slightly 
 * faster than the defaultComparator for Dates and strings, but has not been benchmarked. 
 * Unlike defaultComparator, this comparator doesn't support mixed types correctly, 
 * i.e. use it with `BTree<string>` or `BTree<number>` but not `BTree<string|number>`.
 * 
 * NaN is not supported.
 * 
 * Note: null is treated like 0 when compared with numbers or Date, but in general 
 *   null is not ordered with respect to strings (neither greater nor less), and 
 *   undefined is not ordered with other types.
 */
export function simpleComparator(a: string, b:string): number;
export function simpleComparator(a: number|null, b:number|null): number;
export function simpleComparator(a: Date|null, b:Date|null): number;
export function simpleComparator(a: (number|string)[], b:(number|string)[]): number;
export function simpleComparator(a: any, b: any): number {
  return a > b ? 1 : a < b ? -1 : 0;
};

/**
 * A reasonably fast collection of key-value pairs with a powerful API. 
 * Largely compatible with the standard Map. BTree is a B+ tree data structure,
 * so the collection is sorted by key.
 * 
 * B+ trees tend to use memory more efficiently than hashtables such as the
 * standard Map, especially when the collection contains a large number of 
 * items. However, maintaining the sort order makes them modestly slower: 
 * O(log size) rather than O(1). This B+ tree implementation supports O(1)
 * fast cloning. It also supports freeze(), which can be used to ensure that
 * a BTree is not changed accidentally.
 * 
 * Confusingly, the ES6 Map.forEach(c) method calls c(value,key) instead of
 * c(key,value), in contrast to other methods such as set() and entries()
 * which put the key first. I can only assume that the order was reversed on 
 * the theory that users would usually want to examine values and ignore keys.
 * BTree's forEach() therefore works the same way, but a second method 
 * `.forEachPair((key,value)=>{...})` is provided which sends you the key
 * first and the value second; this method is slightly faster because it is 
 * the "native" for-each method for this class.
 * 
 * Out of the box, BTree supports keys that are numbers, strings, arrays of 
 * numbers/strings, Date, and objects that have a valueOf() method returning a 
 * number or string. Other data types, such as arrays of Date or custom
 * objects, require a custom comparator, which you must pass as the second 
 * argument to the constructor (the first argument is an optional list of 
 * initial items). Symbols cannot be used as keys because they are unordered
 * (one Symbol is never "greater" or "less" than another).
 * 
 * @example
 * Given a {name: string, age: number} object, you can create a tree sorted by
 * name and then by age like this:
 *   
 *     var tree = new BTree(undefined, (a, b) => {
 *       if (a.name > b.name)
 *         return 1; // Return a number >0 when a > b
 *       else if (a.name < b.name)
 *         return -1; // Return a number <0 when a < b
 *       else // names are equal (or incomparable)
 *         return a.age - b.age; // Return >0 when a.age > b.age
 *     });
 * 
 *     tree.set({name:"Bill", age:17}, "happy");
 *     tree.set({name:"Fran", age:40}, "busy & stressed");
 *     tree.set({name:"Bill", age:55}, "recently laid off");
 *     tree.forEachPair((k, v) => {
 *       console.log(`Name: ${k.name} Age: ${k.age} Status: ${v}`);
 *     });
 * 
 * @description
 * The "range" methods (`forEach, forRange, editRange`) will return the number
 * of elements that were scanned. In addition, the callback can return {break:R}
 * to stop early and return R from the outer function.
 * 
 * - TODO: Test performance of preallocating values array at max size
 * - TODO: Add fast initialization when a sorted array is provided to constructor
 * 
 * For more documentation see https://github.com/qwertie/btree-typescript
 *
 * Are you a C# developer? You might like the similar data structures I made for C#: 
 * BDictionary, BList, etc. See http://core.loyc.net/collections/
 * 
 * @author David Piepgrass
 */
export default class BTree<K=any, V=any> implements ISortedMapF<K,V>, ISortedMap<K,V>
{
  private _root: BNode<K, V> = EmptyLeaf as BNode<K,V>;
  _size: number = 0;
  _maxNodeSize: number;

  /**
   * provides a total order over keys (and a strict partial order over the type K)
   * @returns a negative value if a < b, 0 if a === b and a positive value if a > b
   */
  _compare: (a:K, b:K) => number;
  
  /**
   * Initializes an empty B+ tree.
   * @param compare Custom function to compare pairs of elements in the tree.
   *   If not specified, defaultComparator will be used which is valid as long as K extends DefaultComparable.
   * @param entries A set of key-value pairs to initialize the tree
   * @param maxNodeSize Branching factor (maximum items or children per node)
   *   Must be in range 4..256. If undefined or <4 then default is used; if >256 then 256.
   */
  public constructor(entries?: [K,V][], compare?: (a: K, b: K) => number, maxNodeSize?: number) {
    this._maxNodeSize = maxNodeSize! >= 4 ? Math.min(maxNodeSize!, 256) : 32;
    this._compare = compare || defaultComparator as any as (a: K, b: K) => number;
    if (entries)
      this.setPairs(entries);
  }
  
  /////////////////////////////////////////////////////////////////////////////
  // ES6 Map<K,V> methods /////////////////////////////////////////////////////

  /** Gets the number of key-value pairs in the tree. */
  get size() { return this._size; }
  /** Gets the number of key-value pairs in the tree. */
  get length() { return this._size; }
  /** Returns true iff the tree contains no key-value pairs. */
  get isEmpty() { return this._size === 0; }

  /** Releases the tree so that its size is 0. */
  clear() {
    this._root = EmptyLeaf as BNode<K,V>;
    this._size = 0;
  }

  forEach(callback: (v:V, k:K, tree:BTree<K,V>) => void, thisArg?: any): number;

  /** Runs a function for each key-value pair, in order from smallest to 
   *  largest key. For compatibility with ES6 Map, the argument order to
   *  the callback is backwards: value first, then key. Call forEachPair 
   *  instead to receive the key as the first argument.
   * @param thisArg If provided, this parameter is assigned as the `this`
   *        value for each callback.
   * @returns the number of values that were sent to the callback,
   *        or the R value if the callback returned {break:R}. */
  forEach<R=number>(callback: (v:V, k:K, tree:BTree<K,V>) => {break?:R}|void, thisArg?: any): R|number {
    if (thisArg !== undefined)
      callback = callback.bind(thisArg);
    return this.forEachPair((k, v) => callback(v, k, this));
  }

  /** Runs a function for each key-value pair, in order from smallest to 
   *  largest key. The callback can return {break:R} (where R is any value
   *  except undefined) to stop immediately and return R from forEachPair.
   * @param onFound A function that is called for each key-value pair. This 
   *        function can return {break:R} to stop early with result R.
   *        The reason that you must return {break:R} instead of simply R 
   *        itself is for consistency with editRange(), which allows 
   *        multiple actions, not just breaking.
   * @param initialCounter This is the value of the third argument of 
   *        `onFound` the first time it is called. The counter increases 
   *        by one each time `onFound` is called. Default value: 0
   * @returns the number of pairs sent to the callback (plus initialCounter,
   *        if you provided one). If the callback returned {break:R} then
   *        the R value is returned instead. */
  forEachPair<R=number>(callback: (k:K, v:V, counter:number) => {break?:R}|void, initialCounter?: number): R|number {
    var low = this.minKey(), high = this.maxKey();
    return this.forRange(low!, high!, true, callback, initialCounter);
  }

  /**
   * Finds a pair in the tree and returns the associated value.
   * @param defaultValue a value to return if the key was not found.
   * @returns the value, or defaultValue if the key was not found.
   * @description Computational complexity: O(log size)
   */
  get(key: K, defaultValue?: V): V | undefined {
    return this._root.get(key, defaultValue, this);
  }
  
  /**
   * Adds or overwrites a key-value pair in the B+ tree.
   * @param key the key is used to determine the sort order of
   *        data in the tree.
   * @param value data to associate with the key (optional)
   * @param overwrite Whether to overwrite an existing key-value pair 
   *        (default: true). If this is false and there is an existing
   *        key-value pair then this method has no effect.
   * @returns true if a new key-value pair was added.
   * @description Computational complexity: O(log size)
   * Note: when overwriting a previous entry, the key is updated
   * as well as the value. This has no effect unless the new key
   * has data that does not affect its sort order.
   */
  set(key: K, value: V, overwrite?: boolean): boolean { 
    if (this._root.isShared)
      this._root = this._root.clone();
    var result = this._root.set(key, value, overwrite, this);
    if (result === true || result === false)
      return result;
    // Root node has split, so create a new root node.
    this._root = new BNodeInternal<K,V>([this._root, result]);
    return true;
  }

  /**
   * Returns true if the key exists in the B+ tree, false if not.
   * Use get() for best performance; use has() if you need to
   * distinguish between "undefined value" and "key not present".
   * @param key Key to detect
   * @description Computational complexity: O(log size)
   */
  has(key: K): boolean { 
    return this.forRange(key, key, true, undefined) !== 0;
  }

  /**
   * Removes a single key-value pair from the B+ tree.
   * @param key Key to find
   * @returns true if a pair was found and removed, false otherwise.
   * @description Computational complexity: O(log size)
   */
  delete(key: K): boolean {
    return this.editRange(key, key, true, DeleteRange) !== 0;
  }

  /////////////////////////////////////////////////////////////////////////////
  // Clone-mutators ///////////////////////////////////////////////////////////

  /** Returns a copy of the tree with the specified key set (the value is undefined). */
  with(key: K): BTree<K,V|undefined>;
  /** Returns a copy of the tree with the specified key-value pair set. */
  with<V2>(key: K, value: V2, overwrite?: boolean): BTree<K,V|V2>;
  with<V2>(key: K, value?: V2, overwrite?: boolean): BTree<K,V|V2|undefined> {
    let nu = this.clone() as BTree<K,V|V2|undefined>;
    return nu.set(key, value, overwrite) || overwrite ? nu : this;
  }

  /** Returns a copy of the tree with the specified key-value pairs set. */
  withPairs<V2>(pairs: [K,V|V2][], overwrite: boolean): BTree<K,V|V2> {
    let nu = this.clone() as BTree<K,V|V2>;
    return nu.setPairs(pairs, overwrite) !== 0 || overwrite ? nu : this;
  }

  /** Returns a copy of the tree with the specified keys present. 
   *  @param keys The keys to add. If a key is already present in the tree,
   *         neither the existing key nor the existing value is modified.
   *  @param returnThisIfUnchanged if true, returns this if all keys already 
   *  existed. Performance note: due to the architecture of this class, all
   *  node(s) leading to existing keys are cloned even if the collection is
   *  ultimately unchanged.
  */
  withKeys(keys: K[], returnThisIfUnchanged?: boolean): BTree<K,V|undefined> {
    let nu = this.clone() as BTree<K,V|undefined>, changed = false;
    for (var i = 0; i < keys.length; i++)
      changed = nu.set(keys[i], undefined, false) || changed;
    return returnThisIfUnchanged && !changed ? this : nu;
  }

  /** Returns a copy of the tree with the specified key removed. 
   * @param returnThisIfUnchanged if true, returns this if the key didn't exist.
   *  Performance note: due to the architecture of this class, node(s) leading
   *  to where the key would have been stored are cloned even when the key
   *  turns out not to exist and the collection is unchanged.
   */
  without(key: K, returnThisIfUnchanged?: boolean): BTree<K,V> {
    return this.withoutRange(key, key, true, returnThisIfUnchanged);
  }

  /** Returns a copy of the tree with the specified keys removed.
   * @param returnThisIfUnchanged if true, returns this if none of the keys
   *  existed. Performance note: due to the architecture of this class,
   *  node(s) leading to where the key would have been stored are cloned
   *  even when the key turns out not to exist.
   */
  withoutKeys(keys: K[], returnThisIfUnchanged?: boolean): BTree<K,V> {
    let nu = this.clone();
    return nu.deleteKeys(keys) || !returnThisIfUnchanged ? nu : this;
  }

  /** Returns a copy of the tree with the specified range of keys removed. */
  withoutRange(low: K, high: K, includeHigh: boolean, returnThisIfUnchanged?: boolean): BTree<K,V> {
    let nu = this.clone();
    if (nu.deleteRange(low, high, includeHigh) === 0 && returnThisIfUnchanged)
      return this;
    return nu;
  }

  /** Returns a copy of the tree with pairs removed whenever the callback 
   *  function returns false. `where()` is a synonym for this method. */
  filter(callback: (k:K,v:V,counter:number) => boolean, returnThisIfUnchanged?: boolean): BTree<K,V> {
    var nu = this.greedyClone();
    var del: any;
    nu.editAll((k,v,i) => {
      if (!callback(k, v, i)) return del = Delete;
    });
    if (!del && returnThisIfUnchanged)
      return this;
    return nu;
  }

  /** Returns a copy of the tree with all values altered by a callback function. */
  mapValues<R>(callback: (v:V,k:K,counter:number) => R): BTree<K,R> {
    var tmp = {} as {value:R};
    var nu = this.greedyClone();
    nu.editAll((k,v,i) => {
      return tmp.value = callback(v, k, i), tmp as any;
    });
    return nu as any as BTree<K,R>;
  }

  /** Performs a reduce operation like the `reduce` method of `Array`. 
   *  It is used to combine all pairs into a single value, or perform 
   *  conversions. `reduce` is best understood by example. For example,
   *  `tree.reduce((P, pair) => P * pair[0], 1)` multiplies all keys 
   *  together. It means "start with P=1, and for each pair multiply 
   *  it by the key in pair[0]". Another example would be converting 
   *  the tree to a Map (in this example, note that M.set returns M):
   *  
   *  var M = tree.reduce((M, pair) => M.set(pair[0],pair[1]), new Map())
   *  
   *  **Note**: the same array is sent to the callback on every iteration.
   */
  reduce<R>(callback: (previous:R,currentPair:[K,V],counter:number,tree:BTree<K,V>) => R, initialValue: R): R;
  reduce<R>(callback: (previous:R|undefined,currentPair:[K,V],counter:number,tree:BTree<K,V>) => R): R|undefined;
  reduce<R>(callback: (previous:R|undefined,currentPair:[K,V],counter:number,tree:BTree<K,V>) => R, initialValue?: R): R|undefined {
    let i = 0, p = initialValue;
    var it = this.entries(this.minKey(), ReusedArray), next;
    while (!(next = it.next()).done)
      p = callback(p, next.value, i++, this);
    return p;
  }

  /////////////////////////////////////////////////////////////////////////////
  // Iterator methods /////////////////////////////////////////////////////////

  /** Returns an iterator that provides items in order (ascending order if
   *  the collection's comparator uses ascending order, as is the default.)
   *  @param lowestKey First key to be iterated, or undefined to start at
   *         minKey(). If the specified key doesn't exist then iteration
   *         starts at the next higher key (according to the comparator).
   *  @param reusedArray Optional array used repeatedly to store key-value
   *         pairs, to avoid creating a new array on every iteration.
   */
  entries(lowestKey?: K, reusedArray?: (K|V)[]): IterableIterator<[K,V]> {
    var info = this.findPath(lowestKey);
    if (info === undefined) return iterator<[K,V]>();
    var {nodequeue, nodeindex, leaf} = info;
    var state = reusedArray !== undefined ? 1 : 0;
    var i = (lowestKey === undefined ? -1 : leaf.indexOf(lowestKey, 0, this._compare) - 1);

    return iterator<[K,V]>(() => {
      jump: for (;;) {
        switch(state) {
          case 0:
            if (++i < leaf.keys.length)
              return {done: false, value: [leaf.keys[i], leaf.values[i]]};
            state = 2;
            continue;
          case 1:
            if (++i < leaf.keys.length) {
              reusedArray![0] = leaf.keys[i], reusedArray![1] = leaf.values[i];
              return {done: false, value: reusedArray as [K,V]};
            }
            state = 2;
          case 2:
            // Advance to the next leaf node
            for (var level = -1;;) {
              if (++level >= nodequeue.length) {
                state = 3; continue jump;
              }
              if (++nodeindex[level] < nodequeue[level].length)
                break;
            }
            for (; level > 0; level--) {
              nodequeue[level-1] = (nodequeue[level][nodeindex[level]] as BNodeInternal<K,V>).children;
              nodeindex[level-1] = 0;
            }
            leaf = nodequeue[0][nodeindex[0]];
            i = -1;
            state = reusedArray !== undefined ? 1 : 0;
            continue;
          case 3:
            return {done: true, value: undefined};
        }
      }
    });
  }

  /** Returns an iterator that provides items in reversed order.
   *  @param highestKey Key at which to start iterating, or undefined to 
   *         start at maxKey(). If the specified key doesn't exist then iteration
   *         starts at the next lower key (according to the comparator).
   *  @param reusedArray Optional array used repeatedly to store key-value
   *         pairs, to avoid creating a new array on every iteration.
   *  @param skipHighest Iff this flag is true and the highestKey exists in the
   *         collection, the pair matching highestKey is skipped, not iterated.
   */
  entriesReversed(highestKey?: K, reusedArray?: (K|V)[], skipHighest?: boolean): IterableIterator<[K,V]> {
    if (highestKey === undefined) {
      highestKey = this.maxKey();
      skipHighest = undefined;
      if (highestKey === undefined)
        return iterator<[K,V]>(); // collection is empty
    }
    var {nodequeue,nodeindex,leaf} = this.findPath(highestKey) || this.findPath(this.maxKey())!;
    check(!nodequeue[0] || leaf === nodequeue[0][nodeindex[0]], "wat!");
    var i = leaf.indexOf(highestKey, 0, this._compare);
    if (!skipHighest && i < leaf.keys.length && this._compare(leaf.keys[i], highestKey) <= 0)
      i++;
    var state = reusedArray !== undefined ? 1 : 0;

    return iterator<[K,V]>(() => {
      jump: for (;;) {
        switch(state) {
          case 0:
            if (--i >= 0)
              return {done: false, value: [leaf.keys[i], leaf.values[i]]};
            state = 2;
            continue;
          case 1:
            if (--i >= 0) {
              reusedArray![0] = leaf.keys[i], reusedArray![1] = leaf.values[i];
              return {done: false, value: reusedArray as [K,V]};
            }
            state = 2;
          case 2:
            // Advance to the next leaf node
            for (var level = -1;;) {
              if (++level >= nodequeue.length) {
                state = 3; continue jump;
              }
              if (--nodeindex[level] >= 0)
                break;
            }
            for (; level > 0; level--) {
              nodequeue[level-1] = (nodequeue[level][nodeindex[level]] as BNodeInternal<K,V>).children;
              nodeindex[level-1] = nodequeue[level-1].length-1;
            }
            leaf = nodequeue[0][nodeindex[0]];
            i = leaf.keys.length;
            state = reusedArray !== undefined ? 1 : 0;
            continue;
          case 3:
            return {done: true, value: undefined};
        }
      }
    });
  }

  /* Used by entries() and entriesReversed() to prepare to start iterating.
   * It develops a "node queue" for each non-leaf level of the tree.
   * Levels are numbered "bottom-up" so that level 0 is a list of leaf 
   * nodes from a low-level non-leaf node. The queue at a given level L
   * consists of nodequeue[L] which is the children of a BNodeInternal, 
   * and nodeindex[L], the current index within that child list, such
   * such that nodequeue[L-1] === nodequeue[L][nodeindex[L]].children.
   * (However inside this function the order is reversed.)
   */
  private findPath(key?: K): { nodequeue: BNode<K,V>[][], nodeindex: number[], leaf: BNode<K,V> } | undefined
  {
    var nextnode = this._root;
    var nodequeue: BNode<K,V>[][], nodeindex: number[];

    if (nextnode.isLeaf) {
      nodequeue = EmptyArray, nodeindex = EmptyArray; // avoid allocations
    } else {
      nodequeue = [], nodeindex = [];
      for (var d = 0; !nextnode.isLeaf; d++) {
        nodequeue[d] = (nextnode as BNodeInternal<K,V>).children;
        nodeindex[d] = key === undefined ? 0 : nextnode.indexOf(key, 0, this._compare);
        if (nodeindex[d] >= nodequeue[d].length)
          return; // first key > maxKey()
        nextnode = nodequeue[d][nodeindex[d]];
      }
      nodequeue.reverse();
      nodeindex.reverse();
    }
    return {nodequeue, nodeindex, leaf:nextnode};
  }

  /**
   * Computes the differences between `this` and `other`.
   * For efficiency, the diff is returned via invocations of supplied handlers.
   * The computation is optimized for the case in which the two trees have large amounts 
   * of shared data (obtained by calling the `clone` or `with` APIs) and will avoid 
   * any iteration of shared state.
   * The handlers can cause computation to early exit by returning {break: R}.
   * Neither of the collections should be changed during the comparison process (in your callbacks), as this method assumes they will not be mutated.
   * @param other The tree to compute a diff against.
   * @param onlyThis Callback invoked for all keys only present in `this`.
   * @param onlyOther Callback invoked for all keys only present in `other`.
   * @param different Callback invoked for all keys with differing values.
   */
  diffAgainst<R>(
    other: BTree<K, V>,
    onlyThis?: (k: K, v: V) => { break?: R } | void,
    onlyOther?: (k: K, v: V) => { break?: R } | void,
    different?: (k: K, vThis: V, vOther: V) => { break?: R} | void
  ): R | undefined {
    if (other._compare !== this._compare) {
      throw new Error("Tree comparators are not the same.");
    }

    if (this.isEmpty || other.isEmpty) {
      if (this.isEmpty && other.isEmpty)
        return undefined;
      // If one tree is empty, everything will be an onlyThis/onlyOther.
      if (this.isEmpty)
        return onlyOther === undefined ? undefined : BTree.stepToEnd(BTree.makeDiffCursor(other), onlyOther);
      return onlyThis === undefined ? undefined : BTree.stepToEnd(BTree.makeDiffCursor(this), onlyThis);
    }

    // Cursor-based diff algorithm is as follows:
    // - Until neither cursor has navigated to the end of the tree, do the following:
    //  - If the `this` cursor is "behind" the `other` cursor (strictly <, via compare), advance it.
    //  - Otherwise, advance the `other` cursor.
    //  - Any time a cursor is stepped, perform the following:
    //    - If either cursor points to a key/value pair:
    //      - If thisCursor === otherCursor and the values differ, it is a Different.
    //      - If thisCursor > otherCursor and otherCursor is at a key/value pair, it is an OnlyOther.
    //      - If thisCursor < otherCursor and thisCursor is at a key/value pair, it is an OnlyThis as long as the most recent 
    //        cursor step was *not* otherCursor advancing from a tie. The extra condition avoids erroneous OnlyOther calls 
    //        that would occur due to otherCursor being the "leader".
    //    - Otherwise, if both cursors point to nodes, compare them. If they are equal by reference (shared), skip
    //      both cursors to the next node in the walk.
    // - Once one cursor has finished stepping, any remaining steps (if any) are taken and key/value pairs are logged
    //   as OnlyOther (if otherCursor is stepping) or OnlyThis (if thisCursor is stepping).
    // This algorithm gives the critical guarantee that all locations (both nodes and key/value pairs) in both trees that 
    // are identical by value (and possibly by reference) will be visited *at the same time* by the cursors.
    // This removes the possibility of emitting incorrect diffs, as well as allowing for skipping shared nodes.
    const { _compare } = this;
    const thisCursor = BTree.makeDiffCursor(this);
    const otherCursor = BTree.makeDiffCursor(other);
    // It doesn't matter how thisSteppedLast is initialized.
    // Step order is only used when either cursor is at a leaf, and cursors always start at a node.
    let thisSuccess = true, otherSuccess = true, prevCursorOrder = BTree.compare(thisCursor, otherCursor, _compare);
    while (thisSuccess && otherSuccess) {
      const cursorOrder = BTree.compare(thisCursor, otherCursor, _compare);
      const { leaf: thisLeaf, internalSpine: thisInternalSpine, levelIndices: thisLevelIndices } = thisCursor;
      const { leaf: otherLeaf, internalSpine: otherInternalSpine, levelIndices: otherLevelIndices } = otherCursor;
      if (thisLeaf || otherLeaf) {
        // If the cursors were at the same location last step, then there is no work to be done.
        if (prevCursorOrder !== 0) {
          if (cursorOrder === 0) {
            if (thisLeaf && otherLeaf && different) {
              // Equal keys, check for modifications
              const valThis = thisLeaf.values[thisLevelIndices[thisLevelIndices.length - 1]];
              const valOther = otherLeaf.values[otherLevelIndices[otherLevelIndices.length - 1]];
              if (!Object.is(valThis, valOther)) {
                const result = different(thisCursor.currentKey, valThis, valOther);
                if (result && result.break)
                  return result.break;
              }
            }
          } else if (cursorOrder > 0) {
            // If this is the case, we know that either:
            // 1. otherCursor stepped last from a starting position that trailed thisCursor, and is still behind, or
            // 2. thisCursor stepped last and leapfrogged otherCursor
            // Either of these cases is an "only other"
            if (otherLeaf && onlyOther) {
              const otherVal = otherLeaf.values[otherLevelIndices[otherLevelIndices.length - 1]];
              const result = onlyOther(otherCursor.currentKey, otherVal);
              if (result && result.break)
                return result.break;
            }
          } else if (onlyThis) {
            if (thisLeaf && prevCursorOrder !== 0) {
              const valThis = thisLeaf.values[thisLevelIndices[thisLevelIndices.length - 1]];
              const result = onlyThis(thisCursor.currentKey, valThis);
              if (result && result.break)
                return result.break;
            }
          }
        }
      } else if (!thisLeaf && !otherLeaf && cursorOrder === 0) {
        const lastThis = thisInternalSpine.length - 1;
        const lastOther = otherInternalSpine.length - 1;
        const nodeThis = thisInternalSpine[lastThis][thisLevelIndices[lastThis]];
        const nodeOther = otherInternalSpine[lastOther][otherLevelIndices[lastOther]];
        if (nodeOther === nodeThis) {
          prevCursorOrder = 0;
          thisSuccess = BTree.step(thisCursor, true);
          otherSuccess = BTree.step(otherCursor, true);
          continue;
        }
      }
      prevCursorOrder = cursorOrder;
      if (cursorOrder < 0) {
        thisSuccess = BTree.step(thisCursor);
      } else {
        otherSuccess = BTree.step(otherCursor);
      }
    }

    if (thisSuccess && onlyThis)
      return BTree.finishCursorWalk(thisCursor, otherCursor, _compare, onlyThis);
    if (otherSuccess && onlyOther)
      return BTree.finishCursorWalk(otherCursor, thisCursor, _compare, onlyOther);
  }

  ///////////////////////////////////////////////////////////////////////////
  // Helper methods for diffAgainst /////////////////////////////////////////

  private static finishCursorWalk<K, V, R>(
    cursor: DiffCursor<K, V>,
    cursorFinished: DiffCursor<K, V>,
    compareKeys: (a: K, b: K) => number,
    callback: (k: K, v: V) => { break?: R } | void
  ): R | undefined {
    const compared = BTree.compare(cursor, cursorFinished, compareKeys);
    if (compared === 0) {
      if (!BTree.step(cursor))
        return undefined;
    } else if (compared < 0) {
      check(false, "cursor walk terminated early");
    }
    return BTree.stepToEnd(cursor, callback);
  }

  private static stepToEnd<K, V, R>(
    cursor: DiffCursor<K, V>,
    callback: (k: K, v: V) => { break?: R } | void
  ): R | undefined {
    let canStep: boolean = true;
    while (canStep) {
      const { leaf, levelIndices, currentKey } = cursor;
      if (leaf) {
        const value = leaf.values[levelIndices[levelIndices.length - 1]];
        const result = callback(currentKey, value);
        if (result && result.break)
          return result.break;
      }
      canStep = BTree.step(cursor);
    }
    return undefined;
  }

  private static makeDiffCursor<K, V>(tree: BTree<K, V>): DiffCursor<K, V> {
    const { _root, height } = tree;
    return { height: height, internalSpine: [[_root]], levelIndices: [0], leaf: undefined, currentKey: _root.maxKey() };
  }

  /**
   * Advances the cursor to the next step in the walk of its tree.
   * Cursors are walked backwards in sort order, as this allows them to leverage maxKey() in order to be compared in O(1).
   * @param cursor The cursor to step
   * @param stepToNode If true, the cursor will be advanced to the next node (skipping values)
   * @returns true if the step was completed and false if the step would have caused the cursor to move beyond the end of the tree.
   */ 
  private static step<K, V>(cursor: DiffCursor<K, V>, stepToNode?: boolean): boolean {
    const { internalSpine, levelIndices, leaf } = cursor;
    if (stepToNode === true || leaf) {
      const levelsLength = levelIndices.length;
      // Step to the next node only if:
      // - We are explicitly directed to via stepToNode, or
      // - There are no key/value pairs left to step to in this leaf
      if (stepToNode === true || levelIndices[levelsLength - 1] === 0) {
        const spineLength = internalSpine.length;
        // Root is leaf
        if (spineLength === 0)
          return false;
        // Walk back up the tree until we find a new subtree to descend into
        const nodeLevelIndex = spineLength - 1;
        let levelIndexWalkBack = nodeLevelIndex;
        while (levelIndexWalkBack >= 0) {
          if (levelIndices[levelIndexWalkBack] > 0) {
            if (levelIndexWalkBack < levelsLength - 1) {
              // Remove leaf state from cursor
              cursor.leaf = undefined;
              levelIndices.pop();
            }
            // If we walked upwards past any internal node, slice them out
            if (levelIndexWalkBack < nodeLevelIndex)
              cursor.internalSpine = internalSpine.slice(0, levelIndexWalkBack + 1);
            // Move to new internal node
            cursor.currentKey = internalSpine[levelIndexWalkBack][--levelIndices[levelIndexWalkBack]].maxKey();
            return true;
          }
          levelIndexWalkBack--;
        }
        // Cursor is in the far left leaf of the tree, no more nodes to enumerate
        return false;
      } else {
        // Move to new leaf value
        const valueIndex = --levelIndices[levelsLength - 1];
        cursor.currentKey = (leaf as unknown as BNode<K, V>).keys[valueIndex];
        return true;
      }
    } else { // Cursor does not point to a value in a leaf, so move downwards
      const nextLevel = internalSpine.length;
      const currentLevel = nextLevel - 1;
      const node = internalSpine[currentLevel][levelIndices[currentLevel]];
      if (node.isLeaf) {
        // Entering into a leaf. Set the cursor to point at the last key/value pair.
        cursor.leaf = node;
        const valueIndex = levelIndices[nextLevel] = node.values.length - 1;
        cursor.currentKey = node.keys[valueIndex];
      } else {
        const children = (node as BNodeInternal<K,V>).children;
        internalSpine[nextLevel] = children;
        const childIndex = children.length - 1;
        levelIndices[nextLevel] = childIndex;
        cursor.currentKey = children[childIndex].maxKey();
      }
      return true;
    }
  }

  /**
   * Compares the two cursors. Returns a value indicating which cursor is ahead in a walk.
   * Note that cursors are advanced in reverse sorting order.
   */
  private static compare<K, V>(cursorA: DiffCursor<K, V>, cursorB: DiffCursor<K, V>, compareKeys: (a: K, b: K) => number): number {
    const { height: heightA, currentKey: currentKeyA, levelIndices: levelIndicesA } = cursorA;
    const { height: heightB, currentKey: currentKeyB, levelIndices: levelIndicesB } = cursorB;
    // Reverse the comparison order, as cursors are advanced in reverse sorting order
    const keyComparison = compareKeys(currentKeyB, currentKeyA);
    if (keyComparison !== 0) {
      return keyComparison;
    }

    // Normalize depth values relative to the shortest tree.
    // This ensures that concurrent cursor walks of trees of differing heights can reliably land on shared nodes at the same time.
    // To accomplish this, a cursor that is on an internal node at depth D1 with maxKey X is considered "behind" a cursor on an
    // internal node at depth D2 with maxKey Y, when D1 < D2. Thus, always walking the cursor that is "behind" will allow the cursor
    // at shallower depth (but equal maxKey) to "catch up" and land on shared nodes.
    const heightMin = heightA < heightB ? heightA : heightB;
    const depthANormalized = levelIndicesA.length - (heightA - heightMin);
    const depthBNormalized = levelIndicesB.length - (heightB - heightMin);
    return depthANormalized - depthBNormalized;
  }

  // End of helper methods for diffAgainst //////////////////////////////////
  ///////////////////////////////////////////////////////////////////////////

  /** Returns a new iterator for iterating the keys of each pair in ascending order. 
   *  @param firstKey: Minimum key to include in the output. */
  keys(firstKey?: K): IterableIterator<K> {
    var it = this.entries(firstKey, ReusedArray);
    return iterator<K>(() => {
      var n: IteratorResult<any> = it.next();
      if (n.value) n.value = n.value[0];
      return n;
    });
  }
  
  /** Returns a new iterator for iterating the values of each pair in order by key. 
   *  @param firstKey: Minimum key whose associated value is included in the output. */
  values(firstKey?: K): IterableIterator<V> {
    var it = this.entries(firstKey, ReusedArray);
    return iterator<V>(() => {
      var n: IteratorResult<any> = it.next();
      if (n.value) n.value = n.value[1];
      return n;
    });
  }

  /////////////////////////////////////////////////////////////////////////////
  // Additional methods ///////////////////////////////////////////////////////

  /** Returns the maximum number of children/values before nodes will split. */
  get maxNodeSize() {
    return this._maxNodeSize;
  }

  /** Gets the lowest key in the tree. Complexity: O(log size) */
  minKey(): K | undefined { return this._root.minKey(); }
  
  /** Gets the highest key in the tree. Complexity: O(1) */
  maxKey(): K | undefined { return this._root.maxKey(); }

  /** Quickly clones the tree by marking the root node as shared. 
   *  Both copies remain editable. When you modify either copy, any
   *  nodes that are shared (or potentially shared) between the two
   *  copies are cloned so that the changes do not affect other copies.
   *  This is known as copy-on-write behavior, or "lazy copying". */
  clone(): BTree<K,V> {
    this._root.isShared = true;
    var result = new BTree<K,V>(undefined, this._compare, this._maxNodeSize);
    result._root = this._root;
    result._size = this._size;
    return result;
  }

  /** Performs a greedy clone, immediately duplicating any nodes that are 
   *  not currently marked as shared, in order to avoid marking any 
   *  additional nodes as shared.
   *  @param force Clone all nodes, even shared ones.
   */
  greedyClone(force?: boolean): BTree<K,V> {
    var result = new BTree<K,V>(undefined, this._compare, this._maxNodeSize);
    result._root = this._root.greedyClone(force);
    result._size = this._size;
    return result;
  }

  /** Gets an array filled with the contents of the tree, sorted by key */
  toArray(maxLength: number = 0x7FFFFFFF): [K,V][] {
    let min = this.minKey(), max = this.maxKey();
    if (min !== undefined)
      return this.getRange(min, max!, true, maxLength)
    return [];
  }

  /** Gets an array of all keys, sorted */
  keysArray() {
    var results: K[] = [];
    this._root.forRange(this.minKey()!, this.maxKey()!, true, false, this, 0, 
      (k,v) => { results.push(k); });
    return results;
  }
  
  /** Gets an array of all values, sorted by key */
  valuesArray() {
    var results: V[] = [];
    this._root.forRange(this.minKey()!, this.maxKey()!, true, false, this, 0,
      (k,v) => { results.push(v); });
    return results;
  }

  /** Gets a string representing the tree's data based on toArray(). */
  toString() {
    return this.toArray().toString();
  }

  /** Stores a key-value pair only if the key doesn't already exist in the tree. 
   * @returns true if a new key was added
  */
  setIfNotPresent(key: K, value: V): boolean {
    return this.set(key, value, false);
  }

  /** Returns the next pair whose key is larger than the specified key (or undefined if there is none).
   * If key === undefined, this function returns the lowest pair.
   * @param key The key to search for.
   * @param reusedArray Optional array used repeatedly to store key-value pairs, to 
   * avoid creating a new array on every iteration.
   */
  nextHigherPair(key: K|undefined, reusedArray?: [K,V]): [K,V]|undefined {
    reusedArray = reusedArray || ([] as unknown as [K,V]);
    if (key === undefined) {
      return this._root.minPair(reusedArray);
    }
    return this._root.getPairOrNextHigher(key, this._compare, false, reusedArray);
  }
  
  /** Returns the next key larger than the specified key, or undefined if there is none.
   *  Also, nextHigherKey(undefined) returns the lowest key.
   */
  nextHigherKey(key: K|undefined): K|undefined {
    var p = this.nextHigherPair(key, ReusedArray as [K,V]);
    return p && p[0];
  }

  /** Returns the next pair whose key is smaller than the specified key (or undefined if there is none).
   *  If key === undefined, this function returns the highest pair.
   * @param key The key to search for.
   * @param reusedArray Optional array used repeatedly to store key-value pairs, to 
   *        avoid creating a new array each time you call this method.
   */
  nextLowerPair(key: K|undefined, reusedArray?: [K,V]): [K,V]|undefined {
    reusedArray = reusedArray || ([] as unknown as [K,V]);
    if (key === undefined) {
      return this._root.maxPair(reusedArray);
    }
    return this._root.getPairOrNextLower(key, this._compare, false, reusedArray);
  }
  
  /** Returns the next key smaller than the specified key, or undefined if there is none.
   *  Also, nextLowerKey(undefined) returns the highest key.
   */
  nextLowerKey(key: K|undefined): K|undefined {
    var p = this.nextLowerPair(key, ReusedArray as [K,V]);
    return p && p[0];
  }

  /** Returns the key-value pair associated with the supplied key if it exists 
   *  or the pair associated with the next lower pair otherwise. If there is no
   *  next lower pair, undefined is returned.
   * @param key The key to search for.
   * @param reusedArray Optional array used repeatedly to store key-value pairs, to 
   *        avoid creating a new array each time you call this method.
   * */
  getPairOrNextLower(key: K, reusedArray?: [K,V]): [K,V]|undefined {
    return this._root.getPairOrNextLower(key, this._compare, true, reusedArray || ([] as unknown as [K,V]));
  }

  /** Returns the key-value pair associated with the supplied key if it exists 
   *  or the pair associated with the next lower pair otherwise. If there is no
   *  next lower pair, undefined is returned.
   * @param key The key to search for.
   * @param reusedArray Optional array used repeatedly to store key-value pairs, to 
   *        avoid creating a new array each time you call this method.
   * */
  getPairOrNextHigher(key: K, reusedArray?: [K,V]): [K,V]|undefined {
    return this._root.getPairOrNextHigher(key, this._compare, true, reusedArray || ([] as unknown as [K,V]));
  }

  /** Edits the value associated with a key in the tree, if it already exists. 
   * @returns true if the key existed, false if not.
  */
  changeIfPresent(key: K, value: V): boolean { 
    return this.editRange(key, key, true, (k,v) => ({value})) !== 0;
  }

  /**
   * Builds an array of pairs from the specified range of keys, sorted by key.
   * Each returned pair is also an array: pair[0] is the key, pair[1] is the value.
   * @param low The first key in the array will be greater than or equal to `low`.
   * @param high This method returns when a key larger than this is reached.
   * @param includeHigh If the `high` key is present, its pair will be included
   *        in the output if and only if this parameter is true. Note: if the
   *        `low` key is present, it is always included in the output.
   * @param maxLength Length limit. getRange will stop scanning the tree when 
   *                  the array reaches this size.
   * @description Computational complexity: O(result.length + log size)
   */
  getRange(low: K, high: K, includeHigh?: boolean, maxLength: number = 0x3FFFFFF): [K,V][] {
    var results: [K,V][] = [];
    this._root.forRange(low, high, includeHigh, false, this, 0, (k,v) => {
      results.push([k,v])
      return results.length > maxLength ? Break : undefined;
    });
    return results;
  }

  /** Adds all pairs from a list of key-value pairs.
   * @param pairs Pairs to add to this tree. If there are duplicate keys, 
   *        later pairs currently overwrite earlier ones (e.g. [[0,1],[0,7]] 
   *        associates 0 with 7.)
   * @param overwrite Whether to overwrite pairs that already exist (if false,
   *        pairs[i] is ignored when the key pairs[i][0] already exists.)
   * @returns The number of pairs added to the collection.
   * @description Computational complexity: O(pairs.length * log(size + pairs.length))
   */
  setPairs(pairs: [K,V][], overwrite?: boolean): number {
    var added = 0;
    for (var i = 0; i < pairs.length; i++)
      if (this.set(pairs[i][0], pairs[i][1], overwrite))
        added++;
    return added;
  }

  forRange(low: K, high: K, includeHigh: boolean, onFound?: (k:K,v:V,counter:number) => void, initialCounter?: number): number;

  /**
   * Scans the specified range of keys, in ascending order by key.
   * Note: the callback `onFound` must not insert or remove items in the
   * collection. Doing so may cause incorrect data to be sent to the 
   * callback afterward.
   * @param low The first key scanned will be greater than or equal to `low`.
   * @param high Scanning stops when a key larger than this is reached.
   * @param includeHigh If the `high` key is present, `onFound` is called for
   *        that final pair if and only if this parameter is true.
   * @param onFound A function that is called for each key-value pair. This 
   *        function can return {break:R} to stop early with result R.
   * @param initialCounter Initial third argument of onFound. This value 
   *        increases by one each time `onFound` is called. Default: 0
   * @returns The number of values found, or R if the callback returned 
   *        `{break:R}` to stop early.
   * @description Computational complexity: O(number of items scanned + log size)
   */
  forRange<R=number>(low: K, high: K, includeHigh: boolean, onFound?: (k:K,v:V,counter:number) => {break?:R}|void, initialCounter?: number): R|number {
    var r = this._root.forRange(low, high, includeHigh, false, this, initialCounter || 0, onFound);
    return typeof r === "number" ? r : r.break!;
  }

  /**
   * Scans and potentially modifies values for a subsequence of keys.
   * Note: the callback `onFound` should ideally be a pure function. 
   *   Specfically, it must not insert items, call clone(), or change 
   *   the collection except via return value; out-of-band editing may
   *   cause an exception or may cause incorrect data to be sent to
   *   the callback (duplicate or missed items). It must not cause a 
   *   clone() of the collection, otherwise the clone could be modified
   *   by changes requested by the callback.
   * @param low The first key scanned will be greater than or equal to `low`.
   * @param high Scanning stops when a key larger than this is reached.
   * @param includeHigh If the `high` key is present, `onFound` is called for
   *        that final pair if and only if this parameter is true.
   * @param onFound A function that is called for each key-value pair. This 
   *        function can return `{value:v}` to change the value associated 
   *        with the current key, `{delete:true}` to delete the current pair,
   *        `{break:R}` to stop early with result R, or it can return nothing
   *        (undefined or {}) to cause no effect and continue iterating.
   *        `{break:R}` can be combined with one of the other two commands.
   *        The third argument `counter` is the number of items iterated 
   *        previously; it equals 0 when `onFound` is called the first time.
   * @returns The number of values scanned, or R if the callback returned 
   *        `{break:R}` to stop early.
   * @description 
   *   Computational complexity: O(number of items scanned + log size)
   *   Note: if the tree has been cloned with clone(), any shared
   *   nodes are copied before `onFound` is called. This takes O(n) time 
   *   where n is proportional to the amount of shared data scanned.
   */
  editRange<R=V>(low: K, high: K, includeHigh: boolean, onFound: (k:K,v:V,counter:number) => EditRangeResult<V,R>|void, initialCounter?: number): R|number {
    var root = this._root;
    if (root.isShared)
      this._root = root = root.clone();
    try {
      var r = root.forRange(low, high, includeHigh, true, this, initialCounter || 0, onFound);
      return typeof r === "number" ? r : r.break!;
    } finally {
      let isShared;
      while (root.keys.length <= 1 && !root.isLeaf) {
        isShared ||= root.isShared;
        this._root = root = root.keys.length === 0 ? EmptyLeaf :
                    (root as any as BNodeInternal<K,V>).children[0];
      }
      // If any ancestor of the new root was shared, the new root must also be shared
      if (isShared) {
        root.isShared = true;
      }
    }
  }

  /** Same as `editRange` except that the callback is called for all pairs. */
  editAll<R=V>(onFound: (k:K,v:V,counter:number) => EditRangeResult<V,R>|void, initialCounter?: number): R|number {
    return this.editRange(this.minKey()!, this.maxKey()!, true, onFound, initialCounter);
  }

  /**
   * Removes a range of key-value pairs from the B+ tree.
   * @param low The first key scanned will be greater than or equal to `low`.
   * @param high Scanning stops when a key larger than this is reached.
   * @param includeHigh Specifies whether the `high` key, if present, is deleted.
   * @returns The number of key-value pairs that were deleted.
   * @description Computational complexity: O(log size + number of items deleted)
   */
  deleteRange(low: K, high: K, includeHigh: boolean): number {
    return this.editRange(low, high, includeHigh, DeleteRange);
  }

  /** Deletes a series of keys from the collection. */
  deleteKeys(keys: K[]): number {
    for (var i = 0, r = 0; i < keys.length; i++)
      if (this.delete(keys[i]))
        r++;
    return r;
  }

  /** Gets the height of the tree: the number of internal nodes between the 
   *  BTree object and its leaf nodes (zero if there are no internal nodes). */
  get height(): number {
    let node: BNode<K, V> | undefined = this._root;
    let height = -1;
    while (node) {
      height++;
      node = node.isLeaf ? undefined : (node as unknown as BNodeInternal<K, V>).children[0];
    }
    return height;
  }

  /** Makes the object read-only to ensure it is not accidentally modified.
   *  Freezing does not have to be permanent; unfreeze() reverses the effect.
   *  This is accomplished by replacing mutator functions with a function 
   *  that throws an Error. Compared to using a property (e.g. this.isFrozen) 
   *  this implementation gives better performance in non-frozen BTrees.
   */
  freeze() {
    var t = this as any;
    // Note: all other mutators ultimately call set() or editRange() 
    //       so we don't need to override those others.
    t.clear = t.set = t.editRange = function() {
      throw new Error("Attempted to modify a frozen BTree");
    };
  }

  /** Ensures mutations are allowed, reversing the effect of freeze(). */
  unfreeze() {
    // @ts-ignore "The operand of a 'delete' operator must be optional."
    //            (wrong: delete does not affect the prototype.)
    delete this.clear;
    // @ts-ignore
    delete this.set;
    // @ts-ignore
    delete this.editRange;
  }

  /** Returns true if the tree appears to be frozen. */
  get isFrozen() {
    return this.hasOwnProperty('editRange');
  }

  /** Scans the tree for signs of serious bugs (e.g. this.size doesn't match
   *  number of elements, internal nodes not caching max element properly...)
   *  Computational complexity: O(number of nodes), i.e. O(size). This method
   *  skips the most expensive test - whether all keys are sorted - but it
   *  does check that maxKey() of the children of internal nodes are sorted. */
  checkValid() {
    var size = this._root.checkValid(0, this, 0);
    check(size === this.size, "size mismatch: counted ", size, "but stored", this.size);
  }
}

/** A TypeScript helper function that simply returns its argument, typed as 
 *  `ISortedSet<K>` if the BTree implements it, as it does if `V extends undefined`.
 *  If `V` cannot be `undefined`, it returns `unknown` instead. Or at least, that
 *  was the intention, but TypeScript is acting weird and may return `ISortedSet<K>` 
 *  even if `V` can't be `undefined` (discussion: btree-typescript issue #14) */
export function asSet<K,V>(btree: BTree<K,V>): undefined extends V ? ISortedSet<K> : unknown {
  return btree as any;
}

declare const Symbol: any;
if (Symbol && Symbol.iterator) // iterator is equivalent to entries()
  (BTree as any).prototype[Symbol.iterator] = BTree.prototype.entries;
(BTree as any).prototype.where = BTree.prototype.filter;
(BTree as any).prototype.setRange = BTree.prototype.setPairs;
(BTree as any).prototype.add = BTree.prototype.set; // for compatibility with ISetSink<K>

function iterator<T>(next: () => IteratorResult<T> = (() => ({ done:true, value:undefined }))): IterableIterator<T> {
  var result: any = { next };
  if (Symbol && Symbol.iterator)
    result[Symbol.iterator] = function() { return this; };
  return result;
}


/** Leaf node / base class. **************************************************/
class BNode<K,V> {
  // If this is an internal node, _keys[i] is the highest key in children[i].
  keys: K[];
  values: V[];
  // True if this node might be within multiple `BTree`s (or have multiple parents).
  // If so, it must be cloned before being mutated to avoid changing an unrelated tree.
  // This is transitive: if it's true, children are also shared even if `isShared!=true`
  // in those children. (Certain operations will propagate isShared=true to children.)
  isShared: true | undefined;
  get isLeaf() { return (this as any).children === undefined; }
  
  constructor(keys: K[] = [], values?: V[]) {
    this.keys = keys;
    this.values = values || undefVals as any[];
    this.isShared = undefined;
  }

  ///////////////////////////////////////////////////////////////////////////
  // Shared methods /////////////////////////////////////////////////////////

  maxKey() {
    return this.keys[this.keys.length-1];
  }

  // If key not found, returns i^failXor where i is the insertion index.
  // Callers that don't care whether there was a match will set failXor=0.
  indexOf(key: K, failXor: number, cmp: (a:K, b:K) => number): index {
    const keys = this.keys;
    var lo = 0, hi = keys.length, mid = hi >> 1;
    while(lo < hi) {
      var c = cmp(keys[mid], key);
      if (c < 0)
        lo = mid + 1;
      else if (c > 0) // key < keys[mid]
        hi = mid;
      else if (c === 0)
        return mid;
      else {
        // c is NaN or otherwise invalid
        if (key === key) // at least the search key is not NaN
          return keys.length;
        else
          throw new Error("BTree: NaN was used as a key");
      }
      mid = (lo + hi) >> 1;
    }
    return mid ^ failXor;

    // Unrolled version: benchmarks show same speed, not worth using
    /*var i = 1, c: number = 0, sum = 0;
    if (keys.length >= 4) {
      i = 3;
      if (keys.length >= 8) {
        i = 7;
        if (keys.length >= 16) {
          i = 15;
          if (keys.length >= 32) {
            i = 31;
            if (keys.length >= 64) {
              i = 127;
              i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 64 : -64;
              sum += c;
              i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 32 : -32;
              sum += c;
            }
            i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 16 : -16;
            sum += c;
          }
          i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 8 : -8;
          sum += c;
        }
        i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 4 : -4;
        sum += c;
      }
      i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 2 : -2;
      sum += c;
    }
    i += (c = i < keys.length ? cmp(keys[i], key) : 1) < 0 ? 1 : -1;
    c = i < keys.length ? cmp(keys[i], key) : 1;
    sum += c;
    if (c < 0) {
      ++i;
      c = i < keys.length ? cmp(keys[i], key) : 1;
      sum += c;
    }
    if (sum !== sum) {
      if (key === key) // at least the search key is not NaN
        return keys.length ^ failXor;
      else
        throw new Error("BTree: NaN was used as a key");
    }
    return c === 0 ? i : i ^ failXor;*/
  }

  /////////////////////////////////////////////////////////////////////////////
  // Leaf Node: misc //////////////////////////////////////////////////////////

  minKey(): K | undefined {
    return this.keys[0];
  }

  minPair(reusedArray: [K,V]): [K,V] | undefined {
    if (this.keys.length === 0)
      return undefined;
    reusedArray[0] = this.keys[0];
    reusedArray[1] = this.values[0];
    return reusedArray;
  }

  maxPair(reusedArray: [K,V]): [K,V] | undefined {
    if (this.keys.length === 0)
      return undefined;
    const lastIndex = this.keys.length - 1;
    reusedArray[0] = this.keys[lastIndex];
    reusedArray[1] = this.values[lastIndex];
    return reusedArray;
  }

  clone(): BNode<K,V> {
    var v = this.values;
    return new BNode<K,V>(this.keys.slice(0), v === undefVals ? v : v.slice(0));
  }

  greedyClone(force?: boolean): BNode<K,V> {
    return this.isShared && !force ? this : this.clone();
  }

  get(key: K, defaultValue: V|undefined, tree: BTree<K,V>): V|undefined {
    var i = this.indexOf(key, -1, tree._compare);
    return i < 0 ? defaultValue : this.values[i];
  }

  getPairOrNextLower(key: K, compare: (a: K, b: K) => number, inclusive: boolean, reusedArray: [K,V]): [K,V]|undefined {
    var i = this.indexOf(key, -1, compare);
    const indexOrLower = i < 0 ? ~i - 1 : (inclusive ? i : i - 1);
    if (indexOrLower >= 0) {
      reusedArray[0] = this.keys[indexOrLower];
      reusedArray[1] = this.values[indexOrLower];
      return reusedArray;
    }
    return undefined;
  }

  getPairOrNextHigher(key: K, compare: (a: K, b: K) => number, inclusive: boolean, reusedArray: [K,V]): [K,V]|undefined {
    var i = this.indexOf(key, -1, compare);
    const indexOrLower = i < 0 ? ~i : (inclusive ? i : i + 1);
    const keys = this.keys;
    if (indexOrLower < keys.length) {
      reusedArray[0] = keys[indexOrLower];
      reusedArray[1] = this.values[indexOrLower];
      return reusedArray;
    }
    return undefined;
  }

  checkValid(depth: number, tree: BTree<K,V>, baseIndex: number): number {
    var kL = this.keys.length, vL = this.values.length;
    check(this.values === undefVals ? kL <= vL : kL === vL,
      "keys/values length mismatch: depth", depth, "with lengths", kL, vL, "and baseIndex", baseIndex);
    // Note: we don't check for "node too small" because sometimes a node
    // can legitimately have size 1. This occurs if there is a batch 
    // deletion, leaving a node of size 1, and the siblings are full so
    // it can't be merged with adjacent nodes. However, the parent will
    // verify that the average node size is at least half of the maximum.
    check(depth == 0 || kL > 0, "empty leaf at depth", depth, "and baseIndex", baseIndex);
    return kL;
  }

  /////////////////////////////////////////////////////////////////////////////
  // Leaf Node: set & node splitting //////////////////////////////////////////

  set(key: K, value: V, overwrite: boolean|undefined, tree: BTree<K,V>): boolean|BNode<K,V> { 
    var i = this.indexOf(key, -1, tree._compare);
    if (i < 0) {
      // key does not exist yet
      i = ~i;
      tree._size++;
      
      if (this.keys.length < tree._maxNodeSize) {
        return this.insertInLeaf(i, key, value, tree);
      } else {
        // This leaf node is full and must split
        var newRightSibling = this.splitOffRightSide(), target: BNode<K,V> = this;
        if (i > this.keys.length) {
          i -= this.keys.length;
          target = newRightSibling;
        }
        target.insertInLeaf(i, key, value, tree);
        return newRightSibling;
      }
    } else {
      // Key already exists
      if (overwrite !== false) {
        if (value !== undefined)
          this.reifyValues();
        // usually this is a no-op, but some users may wish to edit the key
        this.keys[i] = key;
        this.values[i] = value;
      }
      return false;
    }
  }

  reifyValues() {
    if (this.values === undefVals)
      return this.values = this.values.slice(0, this.keys.length);
    return this.values;
  }

  insertInLeaf(i: index, key: K, value: V, tree: BTree<K,V>) {
    this.keys.splice(i, 0, key);
    if (this.values === undefVals) {
      while (undefVals.length < tree._maxNodeSize)
        undefVals.push(undefined);
      if (value === undefined) {
        return true;
      } else {
        this.values = undefVals.slice(0, this.keys.length - 1);
      }
    }
    this.values.splice(i, 0, value);
    return true;
  }
  
  takeFromRight(rhs: BNode<K,V>) {
    // Reminder: parent node must update its copy of key for this node
    // assert: neither node is shared
    // assert rhs.keys.length > (maxNodeSize/2 && this.keys.length<maxNodeSize)
    var v = this.values;
    if (rhs.values === undefVals) {
      if (v !== undefVals)
        v.push(undefined as any);
    } else {
      v = this.reifyValues();
      v.push(rhs.values.shift()!);
    }
    this.keys.push(rhs.keys.shift()!);
  }

  takeFromLeft(lhs: BNode<K,V>) {
    // Reminder: parent node must update its copy of key for this node
    // assert: neither node is shared
    // assert rhs.keys.length > (maxNodeSize/2 && this.keys.length<maxNodeSize)
    var v = this.values;
    if (lhs.values === undefVals) {
      if (v !== undefVals)
        v.unshift(undefined as any);
    } else {
      v = this.reifyValues();
      v.unshift(lhs.values.pop()!);
    }
    this.keys.unshift(lhs.keys.pop()!);
  }

  splitOffRightSide(): BNode<K,V> {
    // Reminder: parent node must update its copy of key for this node
    var half = this.keys.length >> 1, keys = this.keys.splice(half);
    var values = this.values === undefVals ? undefVals : this.values.splice(half);
    return new BNode<K,V>(keys, values);
  }

  /////////////////////////////////////////////////////////////////////////////
  // Leaf Node: scanning & deletions //////////////////////////////////////////

  forRange<R>(low: K, high: K, includeHigh: boolean|undefined, editMode: boolean, tree: BTree<K,V>, count: number,
              onFound?: (k:K, v:V, counter:number) => EditRangeResult<V,R>|void): EditRangeResult<V,R>|number {
    var cmp = tree._compare;
    var iLow, iHigh;
    if (high === low) {
      if (!includeHigh)
        return count;
      iHigh = (iLow = this.indexOf(low, -1, cmp)) + 1;
      if (iLow < 0)
        return count;
    } else {
      iLow = this.indexOf(low, 0, cmp);
      iHigh = this.indexOf(high, -1, cmp);
      if (iHigh < 0)
        iHigh = ~iHigh;
      else if (includeHigh === true)
        iHigh++;
    }
    var keys = this.keys, values = this.values;
    if (onFound !== undefined) {
      for(var i = iLow; i < iHigh; i++) {
        var key = keys[i];
        var result = onFound(key, values[i], count++);
        if (result !== undefined) {
          if (editMode === true) {
            if (key !== keys[i] || this.isShared === true)
              throw new Error("BTree illegally changed or cloned in editRange");
            if (result.delete) {
              this.keys.splice(i, 1);
              if (this.values !== undefVals)
                this.values.splice(i, 1);
              tree._size--;
              i--;
              iHigh--;
            } else if (result.hasOwnProperty('value')) {
              values![i] = result.value!;
            }
          }
          if (result.break !== undefined)
            return result;
        }
      }
    } else
      count += iHigh - iLow;
    return count;
  }

  /** Adds entire contents of right-hand sibling (rhs is left unchanged) */
  mergeSibling(rhs: BNode<K,V>, _: number) {
    this.keys.push.apply(this.keys, rhs.keys);
    if (this.values === undefVals) {
      if (rhs.values === undefVals)
        return;
      this.values = this.values.slice(0, this.keys.length);
    }
    this.values.push.apply(this.values, rhs.reifyValues());
  }
}

/** Internal node (non-leaf node) ********************************************/
class BNodeInternal<K,V> extends BNode<K,V> {
  // Note: conventionally B+ trees have one fewer key than the number of 
  // children, but I find it easier to keep the array lengths equal: each
  // keys[i] caches the value of children[i].maxKey().
  children: BNode<K,V>[];

  /** 
   * This does not mark `children` as shared, so it is the responsibility of the caller
   * to ensure children are either marked shared, or aren't included in another tree.
   */
  constructor(children: BNode<K,V>[], keys?: K[]) {
    if (!keys) {
      keys = [];
      for (var i = 0; i < children.length; i++)
        keys[i] = children[i].maxKey();
    }
    super(keys);
    this.children = children;
  }

  clone(): BNode<K,V> {
    var children = this.children.slice(0);
    for (var i = 0; i < children.length; i++)
      children[i].isShared = true;
    return new BNodeInternal<K,V>(children, this.keys.slice(0));
  }

  greedyClone(force?: boolean): BNode<K,V> {
    if (this.isShared && !force)
      return this;
    var nu = new BNodeInternal<K,V>(this.children.slice(0), this.keys.slice(0));
    for (var i = 0; i < nu.children.length; i++)
      nu.children[i] = nu.children[i].greedyClone(force);
    return nu;
  }

  minKey() {
    return this.children[0].minKey();
  }

  minPair(reusedArray: [K,V]): [K,V] | undefined {
    return this.children[0].minPair(reusedArray);
  }

  maxPair(reusedArray: [K,V]): [K,V] | undefined {
    return this.children[this.children.length - 1].maxPair(reusedArray);
  }

  get(key: K, defaultValue: V|undefined, tree: BTree<K,V>): V|undefined {
    var i = this.indexOf(key, 0, tree._compare), children = this.children;
    return i < children.length ? children[i].get(key, defaultValue, tree) : undefined;
  }

  getPairOrNextLower(key: K, compare: (a: K, b: K) => number, inclusive: boolean, reusedArray: [K,V]): [K,V]|undefined {
    var i = this.indexOf(key, 0, compare), children = this.children;
    if (i >= children.length)
      return this.maxPair(reusedArray);
    const result = children[i].getPairOrNextLower(key, compare, inclusive, reusedArray);
    if (result === undefined && i > 0) {
      return children[i - 1].maxPair(reusedArray);
    }
    return result;
  }

  getPairOrNextHigher(key: K, compare: (a: K, b: K) => number, inclusive: boolean, reusedArray: [K,V]): [K,V]|undefined {
    var i = this.indexOf(key, 0, compare), children = this.children, length = children.length;
    if (i >= length)
      return undefined;
    const result = children[i].getPairOrNextHigher(key, compare, inclusive, reusedArray);
    if (result === undefined && i < length - 1) {
      return children[i + 1].minPair(reusedArray);
    }
    return result;
  }

  checkValid(depth: number, tree: BTree<K,V>, baseIndex: number): number {
    let kL = this.keys.length, cL = this.children.length;
    check(kL === cL, "keys/children length mismatch: depth", depth, "lengths", kL, cL, "baseIndex", baseIndex);
    check(kL > 1 || depth > 0, "internal node has length", kL, "at depth", depth, "baseIndex", baseIndex);
    let size = 0, c = this.children, k = this.keys, childSize = 0;
    for (var i = 0; i < cL; i++) {
      size += c[i].checkValid(depth + 1, tree, baseIndex + size);
      childSize += c[i].keys.length;
      check(size >= childSize, "wtf", baseIndex); // no way this will ever fail
      check(i === 0 || c[i-1].constructor === c[i].constructor, "type mismatch, baseIndex:", baseIndex);
      if (c[i].maxKey() != k[i])
        check(false, "keys[", i, "] =", k[i], "is wrong, should be ", c[i].maxKey(), "at depth", depth, "baseIndex", baseIndex);
      if (!(i === 0 || tree._compare(k[i-1], k[i]) < 0))
        check(false, "sort violation at depth", depth, "index", i, "keys", k[i-1], k[i]);
    }
    // 2020/08: BTree doesn't always avoid grossly undersized nodes,
    // but AFAIK such nodes are pretty harmless, so accept them.
    let toofew = childSize === 0; // childSize < (tree.maxNodeSize >> 1)*cL;
    if (toofew || childSize > tree.maxNodeSize*cL)
      check(false, toofew ? "too few" : "too many", "children (", childSize, size, ") at depth", depth, "maxNodeSize:", tree.maxNodeSize, "children.length:", cL, "baseIndex:", baseIndex);
    return size;
  }

  /////////////////////////////////////////////////////////////////////////////
  // Internal Node: set & node splitting //////////////////////////////////////

  set(key: K, value: V, overwrite: boolean|undefined, tree: BTree<K,V>): boolean|BNodeInternal<K,V> {
    var c = this.children, max = tree._maxNodeSize, cmp = tree._compare;
    var i = Math.min(this.indexOf(key, 0, cmp), c.length - 1), child = c[i];
    
    if (child.isShared)
      c[i] = child = child.clone();
    if (child.keys.length >= max) {
      // child is full; inserting anything else will cause a split.
      // Shifting an item to the left or right sibling may avoid a split.
      // We can do a shift if the adjacent node is not full and if the
      // current key can still be placed in the same node after the shift.
      var other: BNode<K,V>;
      if (i > 0 && (other = c[i-1]).keys.length < max && cmp(child.keys[0], key) < 0) {
        if (other.isShared)
          c[i-1] = other = other.clone();
        other.takeFromRight(child);
        this.keys[i-1] = other.maxKey();
      } else if ((other = c[i+1]) !== undefined && other.keys.length < max && cmp(child.maxKey(), key) < 0) {
        if (other.isShared)
          c[i+1] = other = other.clone();
        other.takeFromLeft(child);
        this.keys[i] = c[i].maxKey();
      }
    }

    var result = child.set(key, value, overwrite, tree);
    if (result === false)
      return false;
    this.keys[i] = child.maxKey();
    if (result === true)
      return true;

    // The child has split and `result` is a new right child... does it fit?
    if (this.keys.length < max) { // yes
      this.insert(i+1, result);
      return true;
    } else { // no, we must split also
      var newRightSibling = this.splitOffRightSide(), target: BNodeInternal<K,V> = this;
      if (cmp(result.maxKey(), this.maxKey()) > 0) {
        target = newRightSibling;
        i -= this.keys.length;
      }
      target.insert(i+1, result);
      return newRightSibling;
    }
  }

  /** 
   * Inserts `child` at index `i`.
   * This does not mark `child` as shared, so it is the responsibility of the caller
   * to ensure that either child is marked shared, or it is not included in another tree.
   */
  insert(i: index, child: BNode<K,V>) {
    this.children.splice(i, 0, child);
    this.keys.splice(i, 0, child.maxKey());
  }

  /**
   * Split this node.
   * Modifies this to remove the second half of the items, returning a separate node containing them.
   */
  splitOffRightSide() {
    // assert !this.isShared;
    var half = this.children.length >> 1;
    return new BNodeInternal<K,V>(this.children.splice(half), this.keys.splice(half));
  }

  takeFromRight(rhs: BNode<K,V>) {
    // Reminder: parent node must update its copy of key for this node
    // assert: neither node is shared
    // assert rhs.keys.length > (maxNodeSize/2 && this.keys.length<maxNodeSize)
    this.keys.push(rhs.keys.shift()!);
    this.children.push((rhs as BNodeInternal<K,V>).children.shift()!);
  }

  takeFromLeft(lhs: BNode<K,V>) {
    // Reminder: parent node must update its copy of key for this node
    // assert: neither node is shared
    // assert rhs.keys.length > (maxNodeSize/2 && this.keys.length<maxNodeSize)
    this.keys.unshift(lhs.keys.pop()!);
    this.children.unshift((lhs as BNodeInternal<K,V>).children.pop()!);
  }

  /////////////////////////////////////////////////////////////////////////////
  // Internal Node: scanning & deletions //////////////////////////////////////

  // Note: `count` is the next value of the third argument to `onFound`. 
  //       A leaf node's `forRange` function returns a new value for this counter,
  //       unless the operation is to stop early.
  forRange<R>(low: K, high: K, includeHigh: boolean|undefined, editMode: boolean, tree: BTree<K,V>, count: number,
    onFound?: (k:K, v:V, counter:number) => EditRangeResult<V,R>|void): EditRangeResult<V,R>|number
  {
    var cmp = tree._compare;
    var keys = this.keys, children = this.children;
    var iLow = this.indexOf(low, 0, cmp), i = iLow;
    var iHigh = Math.min(high === low ? iLow : this.indexOf(high, 0, cmp), keys.length-1);
    if (!editMode) {
      // Simple case
      for(; i <= iHigh; i++) {
        var result = children[i].forRange(low, high, includeHigh, editMode, tree, count, onFound);
        if (typeof result !== 'number')
          return result;
        count = result;
      }
    } else if (i <= iHigh) {
      try {
        for (; i <= iHigh; i++) {
          if (children[i].isShared)
            children[i] = children[i].clone();
          var result = children[i].forRange(low, high, includeHigh, editMode, tree, count, onFound);
          // Note: if children[i] is empty then keys[i]=undefined.
          //       This is an invalid state, but it is fixed below.
          keys[i] = children[i].maxKey();
          if (typeof result !== 'number')
            return result;
          count = result;
        }
      } finally {
        // Deletions may have occurred, so look for opportunities to merge nodes.
        var half = tree._maxNodeSize >> 1;
        if (iLow > 0)
          iLow--;
        for (i = iHigh; i >= iLow; i--) {
          if (children[i].keys.length <= half) {
            if (children[i].keys.length !== 0) {
              this.tryMerge(i, tree._maxNodeSize);
            } else { // child is empty! delete it!
              keys.splice(i, 1);
              children.splice(i, 1);
            }
          }
        }
        if (children.length !== 0 && children[0].keys.length === 0)
          check(false, "emptiness bug");
      }
    }
    return count;
  }

  /** Merges child i with child i+1 if their combined size is not too large */
  tryMerge(i: index, maxSize: number): boolean {
    var children = this.children;
    if (i >= 0 && i + 1 < children.length) {
      if (children[i].keys.length + children[i+1].keys.length <= maxSize) {
        if (children[i].isShared) // cloned already UNLESS i is outside scan range
          children[i] = children[i].clone();
        children[i].mergeSibling(children[i+1], maxSize);
        children.splice(i + 1, 1);
        this.keys.splice(i + 1, 1);
        this.keys[i] = children[i].maxKey();
        return true;
      }
    }
    return false;
  }

  /**
   * Move children from `rhs` into this.
   * `rhs` must be part of this tree, and be removed from it after this call
   * (otherwise isShared for its children could be incorrect).
   */
  mergeSibling(rhs: BNode<K,V>, maxNodeSize: number) {
    // assert !this.isShared;
    var oldLength = this.keys.length;
    this.keys.push.apply(this.keys, rhs.keys);
    const rhsChildren = (rhs as any as BNodeInternal<K,V>).children;
    this.children.push.apply(this.children, rhsChildren);

    if (rhs.isShared && !this.isShared) {
      // All children of a shared node are implicitly shared, and since their new
      // parent is not shared, they must now be explicitly marked as shared.
      for (var i = 0; i < rhsChildren.length; i++)
        rhsChildren[i].isShared = true;
    }

    // If our children are themselves almost empty due to a mass-delete,
    // they may need to be merged too (but only the oldLength-1 and its
    // right sibling should need this).
    this.tryMerge(oldLength-1, maxNodeSize);
  }
}

/**
 * A walkable pointer into a BTree for computing efficient diffs between trees with shared data.
 * - A cursor points to either a key/value pair (KVP) or a node (which can be either a leaf or an internal node). 
 *    As a consequence, a cursor cannot be created for an empty tree.
 * - A cursor can be walked forwards using `step`. A cursor can be compared to another cursor to 
 *    determine which is ahead in advancement.
 * - A cursor is valid only for the tree it was created from, and only until the first edit made to 
 *    that tree since the cursor's creation.
 * - A cursor contains a key for the current location, which is the maxKey when the cursor points to a node 
 *    and a key corresponding to a value when pointing to a leaf.
 * - Leaf is only populated if the cursor points to a KVP. If this is the case, levelIndices.length === internalSpine.length + 1
 *    and levelIndices[levelIndices.length - 1] is the index of the value.
 */
type DiffCursor<K,V> = { height: number, internalSpine: BNode<K,V>[][], levelIndices: number[], leaf: BNode<K,V> | undefined, currentKey: K };

// Optimization: this array of `undefined`s is used instead of a normal
// array of values in nodes where `undefined` is the only value.
// Its length is extended to max node size on first use; since it can
// be shared between trees with different maximums, its length can only
// increase, never decrease. Its type should be undefined[] but strangely
// TypeScript won't allow the comparison V[] === undefined[]. To prevent
// users from making this array too large, BTree has a maximum node size.
//
// FAQ: undefVals[i] is already undefined, so why increase the array size?
// Reading outside the bounds of an array is relatively slow because it
// has the side effect of scanning the prototype chain.
var undefVals: any[] = [];

const Delete = {delete: true}, DeleteRange = () => Delete;
const Break = {break: true};
const EmptyLeaf = (function() { 
  var n = new BNode<any,any>(); n.isShared = true; return n;
})();
const EmptyArray: any[] = [];
const ReusedArray: any[] = []; // assumed thread-local

function check(fact: boolean, ...args: any[]) {
  if (!fact) {
    args.unshift('B+ tree'); // at beginning of message
    throw new Error(args.join(' '));
  }
}

/** A BTree frozen in the empty state. */
export const EmptyBTree = (() => { let t = new BTree(); t.freeze(); return t; })();