## Binary Search Tree: In-Order Predecessor And Successor

### February 12, 2013

We begin with some necessary infrastructure for handing binary search trees; our trees have only keys, not values, but you can add those yourself:

`(define (tree key lkid rkid) (vector key lkid rkid))`

`(define (key tree) (vector-ref tree 0))`

(define (lkid tree) (vector-ref tree 1))

(define (rkid tree) (vector-ref tree 2))

`(define nil (vector 'nil 'nil 'nil))`

(vector-set! nil 1 nil)

(vector-set! nil 2 nil)

(define (nil? tree) (eqv? tree nil))

`(define (insert lt? t k)`

(cond ((nil? t) (tree k nil nil))

((lt? k (key t)) (tree (key t) (insert lt? (lkid t) k) (rkid t)))

((lt? (key t) k) (tree (key t) (lkid t) (insert lt? (rkid t) k)))

(else t)))

Next we write two functions that are subroutines to the `predecessor`

and `successor`

functions as well as being useful in their own right. The `minimum`

function finds the smallest key in the tree by chasing left sub-trees; the `maximum`

function finds the largest key in the tree by chasing right sub-trees:

`(define (minimum t)`

(cond ((nil? t) #f)

((nil? (lkid t)) (key t))

(else (minimum (lkid t)))))

`(define (maximum t)`

(cond ((nil? t) #f)

((nil? (rkid t)) (key t))

(else (maximum (rkid t)))))

Now we are ready for the `predecessor`

and `successor`

functions. There are three returns and two recursive calls in each function. The first and fourth `cond`

clauses return the saved tree when the recursion reaches a null node because the target key does not exist. The fifth clause returns the minimum/maximum when the search key is found. The other two clauses recur. Note that one of the clauses resets the saved tree as it recurs; the saved tree is the parent of the leftmost/rightmost non-matching node, and is the return value if the search key is found at a leaf of the tree:

`(define (pred lt? t k)`

(define (return t) (if (nil? t) #f (key t)))

(let loop ((t t) (prev nil))

(cond ((nil? t) (return prev))

((lt? (key t) k) (loop (rkid t) t))

((lt? k (key t)) (loop (lkid t) prev))

((nil? (lkid t)) (return prev))

(else (maximum (lkid t))))))

`(define (succ lt? t k)`

(define (return t) (if (nil? t) #f (key t)))

(let loop ((t t) (next nil))

(cond ((nil? t) (return next))

((lt? k (key t)) (loop (lkid t) t))

((lt? (key t) k) (loop (rkid t) next))

((nil? (rkid t)) (return next))

(else (minimum (rkid t))))))

You can see the complete code, along with a small set of examples, at http://programmingpraxis.codepad.org/PkWVHKLi.

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Here’s a version in Racket: Predecessor and successor in a binary search tree

I went a slightly different direction than the solution here with a shared function to return the node containing the value we were searching for and the last left and right branch all at once (the ability to return multiple

`value`

s is always nice). That made the actual`predecessor`

and`successor`

functions somewhat simpler:I also included code for using this to implement a sorting algorithm. Dump everything into a tree, find the

`minimum`

, then repeatedly call`successor`

until you run out of elements. Am I incorrect in thinking that this is actually still`O(n log n)`

, albeit with a higher constant than most? It sorts a million element list only about 3x slower than a naive implementation of`quicksort`

did, but insertion sort (known`O(n^2)`

) didn’t even finish before I got bored.I finally got around to this challenge. Here’s the code for successor in python.