Write The Shortest Program to Calculate Height of a Binary Tree

Jelly, 3 bytes

ŒḊ’

A monadic Link accepting a list representing the tree: [root_value, left_tree, right_tree], where each of left_tree and right_tree are similar structures (empty if need be), which yields the height.

Try it online!

How?

Pretty trivial in Jelly:

ŒḊ’ - Link: list, as described above
ŒḊ  - depth
  ’ - decremented (since leaves are `[value, [], []]`)

Python 2,  35  33 bytes

Thanks to Arnauld for noicing an oversight and saving 4.

f=lambda a:a>[]and-~max(map(f,a))

A recursive function accepting a list representing the tree: [root_value, left_tree, right_tree], where each of left_tree and right_tree are similar structures (empty if need be), which returns the height.

Try it online!

^{Note that [] will return False, but in Python False==0.}

05AB1E, 11 7 5 bytes

Δ€`}N

-4 bytes thanks to @ExpiredData.
-2 bytes thanks to @Grimy.

Input format is similar as the Jelly answer: a list representing the tree: [root_value, left_tree, right_tree], where each of left_tree and right_tree are similar structures (optionally empty). I.e. [2,[7,[2,[],[]],[6,[5,[],[]],[11,[],[]]]],[5,[],[9,[4,[],[]],[]]]] represents the tree from the challenge description.

Try it online or verify a few more test cases.

Explanation:

Δ     # Loop until the (implicit) input-list no longer changes:
  €`  #  Flatten the list one level
}N    # After the loop: push the 0-based index of the loop we just finished
      # (which is output implicitly as result)

Note that although 05AB1E is 0-based, the changes-loop Δ causes the output index to be correct, because it needs an additional iteration to check it no longer changes.