Weight of lowest common ancestor satisfies strong triangle inequality

How do I prove that $$d(x,y)$$, defined as the weight of the lowest common ancestor of $$x,y$$, satisfies the strong triangle inequality:

$$d(x,y) \le \max(d(x,z), d(y,z))$$

How do I even start such a proof?

• Start by trying some examples. Then you'll probably realize that there is a case analysis based on the relative locations of their LCAs. Try all cases.
– D.W.
Mar 13 at 21:25
• How is the weight of a node defined? Mar 14 at 6:12