Trees

• Depth of a node: the length (no. of edges) of the branch to the root
• Height of a tree: the maximal depth (no. of layers - 1)
• Perfect tree: when all leaves are on the same depth h, there are $2^{h+1} - 1$ nodes which all all reachable in $h$ steps
• Complete $n$-tree: each vertex except for for the leafs has $n$ children, in total $\geq 2^h$ nodes (no gaps in array representation)
• Binary tree: every node has at most two children
• Balenced binary tree: the left and right subtrees of every node differ in height by no more than 1
• #edges = #nodes - 1 (because every node execpt for the root has an edge to its parent)

Tree traversal #

On a binary tree, in which: Node (N), Left (L), Right (R):

• Preorder (NLR)
• Postorder (LRN)
• Inorder (LNR)

Expression trees #

In prefix notation:

• Operator before argumentens
• Never needs parentheses!
• Corresponds to preorder traversal

In infix notation:

• The 'usual' way of notating math
• Corresponds to inorder traversal

Tries #

A standard trie $T$ on a collection of words $W$, is a tree with the following properties:

• The root of T is empty, and every other node contains a letter
• The children of a node T contains different letters and are in alphabetical order
• The branches in T from the root correspond exactly with the words in W

Compressed trie: strings are concatenated

Compact trie: nodes store range of indicies referencing positions in word

Let $n$ be the sum of lengths of the words and $m$ be the number of words:

Suffix trie: store only suffixes of substrings since every substring of a string is the prefix of a suffix.

The childs of a Trie can either be implemented has an array or a list. Depending on the sparseness of your Trie either implementation can be more memory efficient. However the lookup times for entries in a linked list are slower because the list has to be traversed.

Heaps #

• Heap property: for each node v, its descendants have a value that is smaller or equal than the value of v.
• enqueue / removeMax in $O(\lg(n))$
• Restoring heap property by:
  • upheap: keep swapping with parent until heap property is maintained.
  • downheap: move node down until it is smaller than its ancestors.
• A heap is efficiently implemented as an array. Since the heap is a complete tree there will be no gaps.

Building a heap is $O(n)$ since the time varies with height of node:

algorithm buildMaxHeap(a, n):
    for i in n/2 .. 1
        fix heap order

Invariant: starts with the leaf nodes as heap, each node is the root of the new heap.

Heapsort: First, construct a max heap in ${O}(n)$ and then keep extracting max element in $O(\lg n)$. Hence the total time complexity is $O(n \lg n)$.

Note that heapsort is not stable.

Binary Search trees (BSTs) #

Search tree property: if $x$ in node $k$ then:

• Everything in the left subtree of $k$ is smaller than $x$
• Everything in the right subtree of $k$ is greater than $x$

Operations: search, add, remove are in ${O}(h)$ where $h$ is the height of the tree $(\lg n)$

Runtimes of BSTs are linear if they are not balanced. Hence, there are several self-balancing BSTs that balance themselves so that the operations stay logarithmic:

• AVL Trees
• Red-Black trees
• Treaps

Rotations #

Rearranges subtrees with the goal to minimize the height of a tree while remaining ordered $O(1)$.

Red-Black tree #

Properties:

• Every node is red or black
• The root is black
• Every leaf is black
• If a node is red, both is children are black
• All simple paths contain the same number of black nodes

Insertion: First, insert a node in color red. Then, recolor and rotate nodes to fix the properties.