Abstract
The 2-height of a planted plane tree is the distance between the root and the second highest leaf (which can be the same as the height). In this note the following theorem is proved: The average 2-height of a planted plane tree with n nodes, considering all such trees to be equally likely, is ( π n ) 1 / 2 − 7 6 + O ( n − 1 / 2 + ε ) , for ε > 0 and n → ∞
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