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COMP 250
Lecture 19
(rooted) trees
- Oct. 23, 2017
Linear Data Structures
Non-Linear Data Structures
linked list array tree graph
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(rooted) trees Oct. 23, 2017 1 Linear Data Structures linked list - - PowerPoint PPT Presentation
(rooted) trees Oct. 23, 2017 1 Linear Data Structures linked list - - PowerPoint PPT Presentation
COMP 250 Lecture 19 (rooted) trees Oct. 23, 2017 1 Linear Data Structures linked list array Non-Linear Data Structures tree graph 2 Tree Example: Organization Hierarchy (McGill) 3
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Tree Example: Organization Hierarchy (McGill)
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person Child 1 Child 2 Child 3
Family Tree (descendents)
Here I ignore spouses (partner).
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person mom dad mom’s mom mom’s dad
Family Tree (ancestors)
dad’s mom dad’s dad
This is an example of a binary tree.
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Tree: UNIX file system
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Tree: Java Classes e.g. GUI
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root
Tree Terminology
node, vertex
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A directed edge is ordered pair: (from, to)
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root
parent child
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Every node except the root is a child, and has exactly one parent.
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For some trees,
- edges are from parent to child
- edges are from child to parent
- edges are both from parent to child and child to
parent.
- edge direction is ignored e.g. common with non-
rooted trees (see final slide of today)
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For some trees,
- edges are from parent to child
- edges are from child to parent
- edges are both from parent to child and child to
parent.
- edge direction is ignored e.g. common with non-
rooted trees (see final slide of today)
Most of definitions today will assume edges are from parent to child.
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Q: If a tree has N nodes, how many edges does it have ?
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Q: If a tree has N nodes, how many edges does it have ? A: N-1 Since every edge is of the form (parent, child), and each node except the root is a child and each child has exactly one parent.
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parent siblings
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Two nodes are siblings if they have the same parent.
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Recursive definition of rooted tree…
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Recursive definition of rooted tree
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A tree T is a finite (& possibly empty) set of 𝑜 nodes such that:
- if 𝑜 > 0 then one of the nodes is the
root r
- if 𝑜 > 1 then the 𝑜 − 1 non-root
nodes are partitioned into (non- empty) subsets T1, T2, …, Tk, each
- f which is a tree (called a subtree”),
and the roots of the subtrees are the children of root r.
This definition does NOT assume edges are from parent to child.
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internal nodes (e.g. file directories) external nodes, leaves
(e.g. files)
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A path in a tree is a sequence of nodes (𝑤1, 𝑤2,…, 𝑤𝑙) such that (𝑤𝑗, 𝑤𝑗+1 ) is an edge. The length of a path is the number of edges in the path (number of nodes – 1)
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A path with just
- ne node (𝑤1)
has length = 0, since it has no edges.
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What is the path length? (2 or 3?)
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ancestor descendent
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Node v is an ancestor of node w if there is a path from v to w. Node w is a descendent of node v.
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root
depth (level) 1 2 3 4 The depth or level of a node is the length of the path from the root to the node.
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depth (level) 1 2 3 4
How to compute depth(v) ?
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depth( v ){ if ( v.parent == null) //root return 0 else return 1 + depth( v.parent ) }
This requires parent links. This is analogous to a ‘prev’ link in a doubly linked list.
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?
1 3 1 1 2 1
The height of a node is the maximum length of a path from that node to a leaf.
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2
1 3 1
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0? 2 1
The height of a node is the maximum length of a path from that node to a leaf.
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? 2
1 3 1
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00 2 1
The height of a node is the maximum length of a path from that node to a leaf.
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2 1 3 1 1 2 1
How to compute height(v) ?
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height(v){ if (v is a leaf) return 0 else{ h = 0 for each child w of v h = max(h, height(w)) return 1 + h } }
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2 1 3 1 1 2 1
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How to implement a tree ?
class TreeNode<T>{ T element; }
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How to implement a tree ?
class TreeNode<T>{ T element; ArrayList< TreeNode<T> > children; TreeNode<T> parent; }
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Another common implementation: ‘first child, next sibling’
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More common implementation: ‘first child, next sibling’ (similar to singly linked lists)
class Tree<T>{ TreeNode<T> root; : // inner class class TreeNode<T>{ T element; TreeNode<T> firstChild; TreeNode<T> nextSibling; : } }
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More common implementation: ‘first child, next sibling’ (similar to singly linked lists)
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class Tree<T>{ TreeNode<T> root; : // inner class class TreeNode<T>{ T element; TreeNode<T> firstChild; TreeNode<T> nextSibling; TreeNode<T> parent; : } }
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A tree of what? Each node has an element (not illustrated on right)
class Tree<T>{ TreeNode<T> root; : // inner class class TreeNode<T>{ T element; TreeNode<T> firstChild; TreeNode<T> nextSibling; : } }
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Exercise
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A tree can be represented using lists, as follows:
tree = root | ( root listOfSubTrees ) listOfSubTrees = tree | tree listOfSubTrees
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Exercise
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A tree can be represented using lists, as follows:
tree = root | ( root listOfSubTrees ) listOfSubTrees = tree | tree listOfSubTrees Draw the tree that corresponds to the following list, where the root elements are single digits.
( 6 ( 2 1 7 ) 3 ( 4 5 ) ( 9 8 0 ) )
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( 6 ( 2 1 7 ) 3 ( 4 5 ) ( 9 8 0 ) )
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6
2 3 4 1 7 5 9 8
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( 6 ( 2 1 7 ) 3 ( 4 5 ) ( 9 8 0 ) )
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6
2 3 4 1 7 5 9 8
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2 3 4 1 7 5 9 8
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ASIDE: Non-rooted trees
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6
2 3 4 1 7 5 9 8
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2 3 4 1 7 5 9 8