Question

Calculate the longest path between two nodes.
The path is in an arch.
Signature of method is:

public static int longestPath(Node n)

In the example binary tree below, it is 4 (going thru 2-3-13-5-2).

This is what I have right now and for the given tree it just returns 0.

public static int longestPath(Node n) {
    if (n != null) {
        longestPath(n, 0);
    }
    return 0;
}
private static int longestPath(Node n, int prevNodePath) {

    if (n != null && n.getLeftSon() != null && n.getRightSon() != null) {
        int currNodePath = countLeftNodes(n.getLeftSon()) + countRightNodes(n.getRightSon());
        int leftLongestPath = countLeftNodes(n.getLeftSon().getLeftSon()) + countRightNodes(n.getLeftSon().getRightSon());
        int rightLongestPath = countLeftNodes(n.getRightSon().getLeftSon()) + countRightNodes(n.getRightSon().getRightSon());

        int longestPath = currNodePath > leftLongestPath ? currNodePath : leftLongestPath;
        longestPath = longestPath > rightLongestPath ? longestPath : rightLongestPath;

        longestPath(n.getLeftSon(), longestPath);
        longestPath(n.getRightSon(), longestPath);

        return longestPath > prevNodePath ? longestPath : prevNodePath;
    }
    return 0;
}
private static int countLeftNodes(Node n) {
    if (n != null) {
        return 1+ countLeftNodes(n.getLeftSon());
    }
    return 0;
}
private static int countRightNodes(Node n) {
    if (n != null) {
        return 1+ countRightNodes(n.getRightSon());
    }
    return 0;
}

I understand that I’m missing a key concept somewhere… My brain goes crazy when I try tracking the flow of execution…
Am I right by saying that by finding the longest path among the root, its left & right nodes and then recurse on its left & right nodes passing them the longest path from previous method invocation and finally (when?) return the longest path, I’m not certain as to how you go about returning it…

Answer 1

Maybe it is just as simple:

public static int longestPath(Node n) {
    if (n != null) {
        return longestPath(n, 0); // forgot return?
    }
    return 0;
}

Its more complicated than one might think at first sight. Consider the following tree:

      1
     / \
    2   3
   / \
  4   5
 / \   \
6   7   8
   / \   \
  9   a   b

In this case, the root node is not even in the longest path (a-7-4-2-5-8-b).

So, what you must do is the following: For each node n you must compute the following:

• compute longest path in left subtree starting with the root of the left subtree (called L)
• compute longest path in right subtree starting with the root of the right subtree (called R)
• compute the longest path in left subtree (not necessarily starting with the root of the left subtree) (called l)
• compute the longest path in right subtree (not necessarily starting with the root of the right subtree) (called r)

Then, decide, which combination maximizes path length:

• L+R+2, i.e. going from a subpath in left subtree to current node and from current node through a subpath in right subtree
• l, i.e. just take the left subtree and exclude the current node (and thus right subtree) from path
• r, i.e. just take the right subtree and exclude the current node (and thus left subtree) from path

So I would do a little hack and for every node not return just a single int, but a triple of integers containing (L+R+2, l, r). The caller then must decide what to do with this result according to the above rules.