# 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)

## Search trees #

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 $\mathcal{O}(h)$

## 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:

Trie | Number of nodes | Memory use |
---|---|---|

Standard trie | $\mathcal{O}(n)$ | $\mathcal{O}(n)$ |

Compressed trie | $\mathcal{O}(m)$ | $\mathcal{O}(n)$ |

Compact trie | $\mathcal{O}(m)$ | $\mathcal{O}(m)$ |

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

### Implementation of a trie data structure in C #

Using an array:

```
#include <stdbool.h>
#define N 26
typedef struct trieNode *trie;
typedef struct trieNode {
bool isEnd;
trie next[N];
} trieNode;
```

Using a linked list:

```
typedef trieNode *trie;
struct trieNode {
char letter;
trie nextVert; // Pointer next child (NULL if none)
trie nextHor; // Pointer to next sibling (NULL if none)
};
```

Depending on your data either implementation can be more memory efficient. However the lookup times for entries in a liked 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 $\mathcal{O}(\lg(n))$- Restoring heap property by:
`upheap`

: keep swapping with parent untill heap property is maintained.`downheap`

: move node down untill it is smaller than its anchestors.

- Efficiently implemented as an array (since the heap is a complete tree)

### Heap Sort #

Construct a heap in $\mathcal{O}(n)$ and then keep removing the min/max in $\mathcal{O}(n\lg(n))$.