Predecessor and successor in a binary search tree

Yesterday’s post from Programming Praxis has us trying to find the predecessor and successor to a given value in a binary search tree. There are actually two general algorithms for doing this, depending on if you have parent pointers or not–but they’re asking for the algorithm without.

The basic idea is that as you recur down the tree looking for a node, you’ll also keep track of the last time you branch in each direction (separately). In their examples, they have the code duplicated for each, but I wrote a single function that returns multiple values so that behavior at least is shared.

In any case, the first thing we need to do is set up a tree structure:

; set up a tree structure
(define-struct tree (value left right) #:transparent)
 
(define (leaf val) (tree val (empty-tree) (empty-tree)))
(define (leaf? tr) (and (tree? tr) (not (tree-left? tr)) (not (tree-right? tr))))
 
(define *empty-tree* (tree (void) (void) (void)))
(define (empty-tree) *empty-tree*)
(define (empty? tr) (eq? tr *empty-tree*))
 
; test if the left node of a tree is non-empty
(define (tree-left? tr)
  (and (tree? tr)
       (not (empty? (tree-left tr)))))
 
; test if the right node of a tree is non-empty
(define (tree-right? tr)
  (and (tree? tr)
       (not (empty? (tree-right tr)))))

Here we have the basic struct along with a few helper functions for making an empty-tree and for testing if we have an empty? subtree, either left or right.

The next step will be to go ahead and write a function for finding the minimum and maximum value in a tree. This doesn’t actually care at all how the values have been stored, it’s just a matter of going left/right until you can’t anymore. Here’s minimum, you can see maximum on GitHub (although it’s essentially the same).

; find the minimum value in a binary search tree
(define (minimum tr)
  (cond
    [(empty? tr) (void)]
    [(tree-left? tr)
     (minimum (tree-left tr))]
    [else
     (tree-value tr)]))

Next, the core of the algorithm, we want to be able to find the node containing a value and the last time we went either left or right. Here’s all of that in one function:

; recur to a value, return: 
;   the last time we went left
;   the node containing the search values
;   the last time we went right
(define (find-nodes tr < val)
  (let loop ([tr tr] [last-left (void)] [last-right (void)])
    (cond
      [(empty? tr) 
       (values last-left tr last-right)]
      [(equal? val (tree-value tr))
       (values last-left tr last-right)]
      [(< val (tree-value tr))
       (loop (tree-left tr) tr last-right)]
      [else
       (loop (tree-right tr) last-left tr)])))

From here, we actually have everything we need to build a successor function:

; get the successor to a value
(define (successor tr < val)
  (define-values (l v r) (find-nodes tr < val))
  (cond
    [(tree-right? v)
     (minimum (tree-right v))]
    [(and (not (void? l)) (not (empty? l)))
     (tree-value l)]))

Basically, we’ll get the last left as l, the tree containing the value we’re looking for as v, and the last right as r. Then, if we can go right from the value, the minimum node there is the successor. If we can’t, that means that we went left at some point getting here. Find the most recent left branch (l) and find it’s value instead.

And that’s it. Here’s predecessor as well:

; get the predecessor to a value
(define (predecessor tr < val)
  (define-values (l v r) (find-nodes tr < val))
  (cond
    [(tree-left? v)
     (maximum (tree-left v))]
    [(and (not (void? r)) (not (empty? r)))
     (tree-value r)]))

One thing that’s interesting about these is that you can actually use them to implement a form of tree sort. Start by finding the minimum value. After that, repeatedly find the successor until you have all of the values:

; sort using a tree, using successor
; this is actually O(n log n) believe it or not
(define (tree-sort < ls)
  (define tr (insert-all (empty-tree) < ls))
  (let loop ([x (minimum tr)])
    (cond
      [(void? x) '()]
      [else
       (cons x (loop (successor tr < x)))])))

It’s actually still O(n log n) (there are n O(log n) insertions to build the tree, a O(log n) search to find the minimum, and a O(log n) search for each value). I feel like it will have a significantly higher constant than other O(n log n) sorts (particularly if the tree isn’t well balanced), but it’s surprisingly effective (at least experimentally on the same order as quicksort):

> (define ls (shuffle (range 1000000)))
> (time (quicksort < ls))
cpu time: 13073 real time: 13113 gc time: 6739
...
> (time (tree-sort < ls))
cpu time: 56426 real time: 56795 gc time: 18085
...
> (time (insertion-sort < ls))
; (Stopped after five minutes)

And that’s it. It’s a bit differently organized than the code that Programming Praxis posted as their solution, but I think it’s still pretty straight forward.

If you’d like to check out the entire source, you can do so on GitHub:
- BST predecessor / successor source

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