1. Trees Definitions Implementation Traversals K-ary Trees
Read Weiss, 4.1 – 4.2
Implementation
Nodes and Links
One Arrays
Three Arrays
Traversals
Preorder, Inorder, Postorder
K-ary Trees
Converting trees: k-ary ↔ binary

2. Definitions Nodes Edges Root node Interior node Leaf Parent Children
Ancestor / Proper ancestor
Descendant / Proper descendant
Level of a node
Height of a node / tree
Degree of a node / tree
Depth of node
Path
Acyclic graph

3. a j b k c g m d l i h f e descendants of “a”
root, height=4, depth=level=0
degree=1
proper descendants of “a” j b k c
interior node, height=2, depth=level=2
degree=2 g m d l i h f e
degree=0
leaf, height=0, depth=level=4
degree of tree = 2
height of tree = 4

4. Implementing a Tree Nodes and Links A B C D Node { Object value;
Node lchild;
Node rchild;
} // Node A B C ▲ ▲ ▲ D
▲=null link ▲ ▲

5. Implementing a Tree One array A B C D A: 0 1 2 3 4 5 6 7 8 9
A[1] is root
lchild of A[i] is A[2i]
rchild of A[i] is A[2i+1]
“-” means array element
is null / not used
A[0] not used as a node
A[0] may be used to hold
general info (e.g.,
number of nodes in tree) A B C ▲ ▲ ▲ D
▲=null link ▲ ▲

6. Implementing a Tree Three array A B C D ▲ ▲=null link root = 4
freelist = 7
null = “/” = -1
Object value: D A - B - C -
int lchild: / / /
int rchild: / / / / 8 / / / / /

7. Traversals Preorder N L R preorder (Node t) if (t == null) return;
visit (t.value());
preorder (t.lchild());
preorder (t.rchild());
} // preorder

8. Traversals Inorder L N R inorder (Node t) if (t == null) return;
inorder (t.lchild());
visit (t.value());
inorder (t.rchild());
} // inorder

9. Traversals Postorder L R N postorder (Node t) if (t == null) return;
postorder (t.lchild());
postorder (t.rchild());
visit (t.value());
} // postorder

10. a j b k c g m d l i h f e preorder: a j k m l b c g i h d f e
inorder: m k l j a b i g h c f d e
postorder: m l k j i h g f e d c b a

11. K-ary Trees a q c e f b d n m i p g k j degree of tree = 4
degree of nodes f and n = 3
height of tree = 3
depth=level of m = 2

12. K-ary Tree => Binary Treeq c e f b d n m i p
K-ary Binary
root root
leftmost child left child
right sibling right child g k j

13. Traversals K-ary Tree Binary Tree n g e j k m d c b f i p q a n g e j
Preorder:
Inorder:
Postorder:
Preorder:
Inorder:
Postorder: