Relationship between height and size of a perfect binary tree

1

In the exercise for Tree.isPerfect() Mosh hints to think about the relationship stated in the title.
Based on the code provided in his solution I’ve surmised that the relationship is size = (2 ^ height + 1) - 1.

public boolean isPerfect() {
    return size() == (Math.pow(2, height() + 1) - 1);
  }

However when checking on the web I’ve found that the relationship is stated as size = 2 ^ height, (since height = log size). Can anyone shed some light as to why Mosh’s relationship is slightly different?

Also I’d like to shamelessly plug the DSA Study Group. We are only a few beginners (myself included) as of now so I still vouched to ask the broader forum for feedback. Feel free to check out my other post for the discord link.

2

After enough searching I eventually found the answer.
At a given height of a tree the max nodes is equal to 2 ^ height.
However the max nodes in the entire tree is equal to (2 ^ height + 1) - 1.