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Tree
A tree is a nonempty collection of vertices(nodes) and edges that satisfies certain requirements.
- a vertex (a node) : a simple object that can have a name and can carry other associated information.
- an edge : a connection between two vertices.
- a path : a connected sequence of zero or more edges.
Definitions:
- root : one distinguished node.
- a parent : one node above a node.
- children : the nodes below a node(a parent node).
- siblings : nodes with the same parent.
- ancestors of a node d : the nodes on the path from d to the root.
- If a is an ancestor of d, then d is a descendant of a.
- The length of a path : the number of edges in the path.
- The depth of a node n : the length of the path from n to the root. (The depth of the root is zero)
- The height of a node n : the length of the path from n to its deepest descendant. (The height of a leaf node is zero)
- The height of a tree : the depth of its deepest node = height of the root
- The subtree
- A binary tree : a tree in which no node has more than two children.
- The leaves or terminal nodes : The nodes with no children
- The level of a node : the number of nodes on the path from the node to the root (not including itself).
The following figure is a sample of a tree.
- root : the node E.
- a parent of the nodes M, P, L and E : the node T
- children of the node A : the nodes A and S
- The siblings of the node R : A and E
- The ancestor nodes of the node S : A and E
- The descendant nodes of R : T, M, P, L, E
- The depth of S : 2
- The height of the tree : 3
- The leaves (terminal nodes) : A, S, M, P, L, E, E
- The level of T : 2
Binary Tree
O. Full binary tree
: a tree in which internal nodes completely fill every level except possibly the last.
O Complete binary tree
: a full binary tree where the internal nodes on the bottom level all appear to the left of the external nodes on that level.
The properties of tree :
- There is exactly one path connecting any two nodes in a tree
- A tree with N nodes has N-1 edges.
- A binary tree with N internal nodes has N+1 external nodes
- a vertex (a node) : a simple object that can have a name and can carry other associated information.
- an edge : a connection between two vertices.
- a path : a connected sequence of zero or more edges.
Definitions:
- root : one distinguished node.
- a parent : one node above a node.
- children : the nodes below a node(a parent node).
- siblings : nodes with the same parent.
- ancestors of a node d : the nodes on the path from d to the root.
- If a is an ancestor of d, then d is a descendant of a.
- The length of a path : the number of edges in the path.
- The depth of a node n : the length of the path from n to the root. (The depth of the root is zero)
- The height of a node n : the length of the path from n to its deepest descendant. (The height of a leaf node is zero)
- The height of a tree : the depth of its deepest node = height of the root
- The subtree
- A binary tree : a tree in which no node has more than two children.
- The leaves or terminal nodes : The nodes with no children
- The level of a node : the number of nodes on the path from the node to the root (not including itself).
The following figure is a sample of a tree.
- root : the node E.
- a parent of the nodes M, P, L and E : the node T
- children of the node A : the nodes A and S
- The siblings of the node R : A and E
- The ancestor nodes of the node S : A and E
- The descendant nodes of R : T, M, P, L, E
- The depth of S : 2
- The height of the tree : 3
- The leaves (terminal nodes) : A, S, M, P, L, E, E
- The level of T : 2
Binary Tree
O. Full binary tree
: a tree in which internal nodes completely fill every level except possibly the last.
O Complete binary tree
: a full binary tree where the internal nodes on the bottom level all appear to the left of the external nodes on that level.
The properties of tree :
- There is exactly one path connecting any two nodes in a tree
- A tree with N nodes has N-1 edges.
- A binary tree with N internal nodes has N+1 external nodes
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