Height of a tree with only one node

Question

According to Wikipedia,

The height of a tree is the length of the path from the root to the deepest node in the tree. A (rooted) tree with only one node (the root) has a height of zero (or one).

I dont get it - is it zero or one (or both)?

Solution 1

It just an assuption you make for the recursive description of the height of a binary tree. You can consider a tree composed by just a node either with 0 height or with 1 height.

If you really want to think about it somehow you can think that

• it's 0 if you consider the height as a edge count (so that a single node doesn't have any edge, hence 0)
• it's 1 if you consider the height as a node count (so that a single node counts as 1)

This is just to describe how much height the smallest tree has, then in any case whenever you add a descending node you will add also a related edge so it will increase accordingly.

In the example provided in wikipedia:

This tree can have height 4 (nodes) or 3 (edges). It depends if you are counting it by edges or by nodes.

Solution 2

One advantage of using a node count rather than an edge count is that it distinguishes the empty case (zero nodes, and node level) from the minimal case (one node, and a node level of one). In some cases, an empty tree will not be meaningful, but in other cases an empty try will be perfectly legitimate.

Solution 3

Depends on convention. There isn't a "right" answer here. I was taught it's 1. But zero is just as correct.

Solution 4

I my opinion, Height of one root node should be 0. It makes practical sense as 2^height is also providing you with the number of nodes at that level.

Solution 5

Assuming you are calculating the height in a recursive manner in the node class I would do this to return the height without including height of the root (java code):

int height(){
    int leftHeight = 0;
    int rightHeight = 0;
    if(left != null)
        leftHeight =+ left.height() + 1;
    if(right != null)
        rightHeight =+ right.height() + 1;
    return Math.max(leftHeight, rightHeight);
}

if you want to include the height of the root, then I would do this:

int height(){
    int leftHeight = 0;
    int rightHeight = 0;
    if(left != null)
        leftHeight =+ left.height();
    if(right != null)
        rightHeight =+ right.height();
    return Math.max(leftHeight, rightHeight) + 1;
}

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Comments

[akshitmahajan]

Best answer can be found at the following link: stackoverflow.com/questions/2597637/…

[Snowman]

Oh ok I see. So theres no separate terms to refer to height of nodes and height of edges individually?

[Jack]

No, there isn't.. the height of a tree is measured as the path length from the root to the deepest node. A path is composed by edges and nodes, and specifically if the path has n edges then it has n+1 nodes (this should be quite trivial), that's why you can have to different base cases: a path composed by just a node has 0 edges but 1 node.

[supercat]

One could argue the opposite, though. If height is defined inclusively, (2^height)-1 is the maximum size of tree for a given height.