5 minute read

1. Binary search recap

algorithm BinarySearch(A, key, start, end):
    if start < end then
        mid := floor((start + end) / 2)
        if key == A[mid] then
            return mid
        if key < A[mid] then
            BinarySearch(A, key, start, mid-1)
        if key > A[mid] then
            BinarySearch(A, key, mid+1, end)

f6f13a3042593211f021b220fd88a728.png => Binary search takes Θ(log~2~n) time

2. Intuition

2.1 Components of a BST

- Two values: `key` and `satellite data`
- Three pointers: `left`, `right`, `parent`  ![ccf68e7ad840ffd79b66e50d142fa114.png](http://hongyu.nl/_resources//25d79aa449f64240b8843edce1b9400b.png)

2.2 BST property



Tips: Be careful that a BST is NOT an almost complete binary tree!!!


3. BST operations

3.1 Searching

TREE-SEARCH: returns a pointer to a node with key k if one exists; otherwise, it returns NIL. The procedure begins its search at the root and traces a simple path downward in the tree. For each node x it encounters, it compares the key k with x.key

{- @x: starting node
 - @k: target key
-- recursive version --
algorithm TreeSearch(x, k) 
    if x == NIL or x.key == k then
        return x
    if k < x.key then
        TreeSearch(x.left, k)
        TreeSearch(x.right, k)

-- iterative version --
algorithm TreeSearch(x, k):
    While x != NIL and x.key != k do
        if k < x.key then
            x = x.left
            x = x.right
    return x

Tip: in TreeSearch(), the sequence of nodes encountered forms a simple path downward from the root.

3.2 Minimum and Maximum:

We can always find an element in a binary search tree whose key is a minimum by following left child pointers from the root until we encounter a NIL —— CRLS

-- we assume that the input node is non-NIL --
algorithm TreeMinimum(x):
    While x.left != NIL do
        x = x.left
    return x

algorithm TreeMaximum(x):
    While x.right != NIL do
        x = x.right
    return x

3.3 Successor and Predecessor

The structure of a binary search tree allows us to determine the successor/predecessor of a node without ever comparing keys. —— CRLS

  • Successor:

    If all keys are distinct, the successor of a node x is the node with the smallest key greater than x.key

9470a449bfd93443a22df3bb1364e081.png In the above diagram, the inorder successor of 8 is 10, the inorder successor of 10 is 12 and inorder successor of 14 is 20.

  • Predecessor:

    If all keys are distinct, the predecessor of a node x is the node with the greatest key smaller than x.key

algorithm TreeSuccessor(x):
-- there are two cases --
{-  1. node x has the right subtree:
 - find the smallest node in its right subtree 
    if x.right != NIL then 
        return TreeMinimum(x.right)
{-  2. node x doesn't have right subtree:
 - find the *lowest* ancestor of x whose left child 
 - is x. 
    y := x.parent
    While y != NIL and x == y.right do
    --> (once x is no longer the right child of y, x > y)
        x = y
        y = y.parent

algorithm TreePredecessor(x):
    -- the opposite --

Final notes:

  • Recall that every node is its ancestor.
  • The worst-case relies on the height of the tree (height matters).

3.4 Insertion

BST insertion is nothing but fnding a correct leaf place (NIL) for the newly inserted node:

{-  Assume keys are distinct and n is the new node that need inserting 
 - with z.key = v and z.left == z.right == NIL 
algorithm TreeInsert(T, n):
    trailing := NIL --> y (in the book) is the trailing pointer that represnts the parent of x
    target   := T.root
    While target != NIL do --> Newly inserted node is always a leaf so we need to find a proper NIL for it
        trailing := target
        if n.key < target.key then 
            target = target.left
            target = target.right
    n.parent := traling --> this is where the trailing pointer comes into play (set the parent pointer of the new node)
    if trailing == NIL then --> empty tree
        T.root = n
    else if n.key < trailing.key then
        trailing.left = n
        trailing.right = n

[Question]: why we need trailing pointer?

… This NIL occupies the position where we wish to place the input item n. We need the trailing pointer y because by the time we find the NIL where n belongs, the search has proceeded one step beyond the node that needs to be changed. —— CLRS

By saying “one step beyond”, it means we cannot use a NIL node to access its parent node that is needed to set n.parent (like a singlely linked-list). In other words, we use the trailing pointer to tack the parent when performing searching.

3.5 Deletion

There are 3 cases: 600345769f0a5d53c23d668e3bf2ab6d.png

Sum up: 9cd6dfeb420ae5ddfe420bdccc7d61f6.png (say z is the node that is wanted to be deleted) If z has:

  • one child / single child: replace the deleted node by NIL or its child directly; -> situation: (a), (b)
  • two children:
    1. first, we need to find z’s successor node y which must be in the right subtree of z; (tips: y has no left child <- PBC see my book notes)
    2. elevate y’s child to take y’s position
    3. repace z by y and give y all z’s children

This eventually boils down to find the successor of the removed node and transplant it.

{-- TRANSPLANT replaces one subtree as a child of its
  - parent with another subtree. When TRANSPLANT replaces
  - the subtree rooted at node 'old' with the subtree
  - rooted at node 'new', node 'old'’s parent becomes node
  - 'new'’s parent, and 'old'’s parent ends up having 
  - 'new' as its appropriate child.
algorithm Transplant(T, old, new):
    -- 'old' is the root --
    if old.parent == NIL then --> Why not old == T.root? 
        T.root = new
    -- set parent node --
    else if old == old.parent.left then
        old.parent.left = new
        old.parent.right = new
    -- set parent pointer --
    if new != NIL then
        new.parent = old.parent

algorithm TreeDelete(T, z):
    -- case (a), (b): single child --
    if z.left == NIL then
        Transplant(z, z.right) 
    else if z.right == NIL then
        Transplant(z, z.left) --> replace directly (assume all subtrees satisfy BST property)
    -- case (c), (d): two children --
        successor := TreeMinimum(z.right) --> y 
        if successor.parent != z then
        -- case d (prepare successor) --
            Transplant(T, successor, successor.right)
            -- update right child --
            successor.right = z.right 
            successor.right.parent = successor
    -- case c, d (replace z by its successor) --
        Transplant(T, z, successor)
        -- update left child --
        successor.left = z.left
        successor.left.parent = successor

Tips: the most time-consuming procedure lies in the TreeMinimum() which may take Θ(logn) time.

4. Time Complexity

  1. All TreeSearch() TreeMinimum() TreeMaximum() TreeSuccessor() TreePredeseccor() take O(h) time on a binary search tree of height h.

  2. For TreeDelete(), Tranplant() takes constant time only TreeMinimun() takes O(h) time, so both TreeInsert() and TreeDelete() take O(h) time on a binary search tree of height h.


5. Intermezzo: How many BT with n nodes exist?


first Catalan numbers: 1; 1; 2; 5; 14; 42; 132; 429; 1430; 4862; 16796; 58786; 208012; 742900; 2674440; 9694845; 35357670; 129644790; 477638700; 1767263190; 6564120420; 24466267020; 91482563640; 343059613650; 1289904147324; 4861946401452; …

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