TreesUC Berkeley

Fall 2004, E77http://jagger.me.berkeley.edu/~pack/e77

Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to

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Trees

Sciences

OrganicMolecular

MathPhysicsChemistryBiology

Quantum Algebra

What is a tree?

A data structure to represent hierarchical or sorted data

Trees: terminology

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Node: any item of the tree

Root Node: initial item of tree

Parent: the parent of F is C

Child: E is a child of B

Leaf Node: A node with no children.

Subtree

A tree in its own right

Ancestors (parent, parent of parent, etc.)

Descendents (children, children of children, etc)

Siblings (children with same parent)

Trees: storing with a struct

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T(1).Data = {’A’ [12 5]};T(2).Data = {’B’ [01 4]};T(3).Data = {’C’ [-7 3]};T(4).Data = {’D’ [41 0]};T(5).Data = {’E’ [.7 9]};T(6).Data = {’F’ [21 2]};T(7).Data = {’G’ [14 1]};T(8).Data = {’H’ [06 6]};T(9).Data = {’I’ [97 4]};T(1).Children = [2 3];T(2).Children = [4 5];T(3).Children = [6 7 8];T(4).Children = [];T(5).Children = [];T(6).Children = [];T(7).Children = 9;T(8).Children = [];T(9).Children = [];

In this example, the data at each node is a cell array, containing the node name and a 1-by-2 array of numerical information

Binary Tree

If every node has 0, 1 or 2 children, then the tree is called a binary tree.

The children of a node in a binary tree are referred to as Left and Right. A

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Binary Search Trees:

Used to store data that can be “sorted”, ie., a notion of ≤ exists for the data.

The data at each node must have the property that Di≤Dk or Dk≤Di.

Moreover, if D1≤D2 and D2≤D3 then it must be that D1≤D3.

This gives efficient search routines. More in Weeks 11, 12 and 13.

Traversing a Tree

An elementary and common operation on a tree is to start at the root node, and

–traverse the tree (ie., visit every node), while…

–performing some operation at each node (using, for example, the Data at the node)

Recursive functions serve this purpose well.

Two traversals: PreOrder and PostOrder

InOrder for Binary trees

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PreOrder traversal of a Tree

function preordop(Tree,Node)

% First, do operation using Data% at the given Node of the TreeOperation(Tree(Node).Data);

% Then loop through all Children,% calling preordop recursivelyfor i=Tree(Node).Children preordop(Tree,i);end

Note: As written here, based on Matlab

syntax, the for loop requires that the children nodes be listed in a row vector.

In this example, Operation represents any desired function to act on the data

Preorder traversal of a Tree

function poprint(T,Node)disp(T(Node).Name)poprint(T,T(Node).Kids(1))…poprint(T,T(Node).Kids(end))

>> poprint(Tree,1)

At A, print A, do same A’s kidsAt B, print B, do same for B’s kids

At D, print D.At E, print E.

At C, print C, do same for C’s kidsAt F, print FAt G, print G, do same for G’s kids

At I, print I. At H, print H

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A B ED C F IG H

Post-order traversal of a Tree

function postordop(Tree,Node)

% First loop through all Children,% calling postordop recursivelyfor i=Tree(Node).Children postordop(Tree,i);end

% Then do operation using the Data% at the given Node of the TreeOperation(Tree(Node).Data);

In-order traversal of a Binary Tree

function inordop(BTree,Node)

% First call inordop recursively% on the “left” childinordop(Tree,Tree(Node).Left);

% Do operation using the Data% at the given Node of the TreeOperation(Tree(Node).Data);

% Finally call inordop recursively% on the “right” childinordop(Tree,Tree(Node)).Right);

InOrder A

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function inordop(BTree,Node)

% First call inordop% on the “left” childinordop(Tree,Tree(Node).Left);

% Do operation using the Data% at the given Node of the TreeOperation(Tree(Node).Data);

% Finally call inordop recursively% on the “right” childinordop(Tree,Tree(Node)).Right);

D, B, E, A, F, C, G