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Tree Traversal Orderings
Level-Order Traversal. Visit top-to-bottom, left-to-right (like reading in English): DBFACEG
Depth-First Traversals. Traverse deep nodes ( A , C , E , G ) before shallow ones ( D , B , F ). Note: “Traversing” a node is different than “visiting” a node. 3 types: Preorder , Inorder , Postorder.
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A C
B
D
E
F
G
Depth-First Traversals
Preorder Traversal. “Visit” a node, then traverse its children.
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A C
B
D
E
F
G
preOrder(BSTNode x) { if (x == null) return; print(x.key) preOrder(x.left) preOrder(x.right) }
D B A C F E G
Depth-First Traversals
Inorder Traversal. Traverse left child, “visit”, then traverse right child.
A C
B
D
E
F
G
inOrder(BSTNode x) { if (x == null) return; inOrder(x.left) print(x.key) inOrder(x.right) }
A B C D E F G
Depth-First Traversals
Postorder Traversal. Traverse left, traverse right, then “visit.”
A C
B
D
E
F
G
postOrder(BSTNode x) { if (x == null) return; postOrder(x.left) postOrder(x.right) print(x.key) }
Q
Depth-First Traversals: Visual Trick (for humans)
First, trace a path around the graph from the top going counter-clockwise. Preorder. “Visit” when passing the left. Inorder. “Visit” when passing the bottom. Postorder. “Visit” when passing the right.
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A C
B
D
E
F
G
A C B E G F D
Alternate Tree Definition
Tree. Consists of a set of nodes and a set of edges that connect those nodes. Invariant. There is exactly one path between any two nodes.
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Graph Definition
Graph. Consists of a set of nodes and a set of zero or more edges. Each edge connects any two nodes. Not all nodes need to be connected.
Simple Graph Definition
Simple Graph. A graph with no self-loops and no parallel edges. Unless otherwise stated, all graphs in this course are simple graphs.
Self-loop
Parallel
connected(s, t):
● Mark s.
● Does s == t? If so, return true.
● Otherwise, if connected(v, t) for any unmarked neighbor v of s, return true.
● Return false.
s-t Connectivity
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2
3
4
5
6
7 8
0 s
t
DepthFirstPaths Demo
Goal: Find a path from s to every other reachable vertex, visiting each vertex at
most once. dfs(v) is as follows:
● Mark v.
● For each unmarked adjacent vertex w:
○ set edgeTo[w] = v.
○ dfs(w)
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2
3
4
5
6
7 8
0 s
marked edgeTo
0 F - 1 F - 2 F - 3 F - 4 F - 5 F - 6 F - 7 F - 8 F - (^) Order of dfs returns:
Order of dfs calls: 0
Start by calling dfs(0).
