Any sequence x = ( x k ) k = 1 ∞ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let H n ( x ) be the height of the tree generated by x 1 , ⋯ , x n . Obviously log n log 2 - 1 ≤ H n ( x ) ≤ n - 1 . If the sequences x are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists c > 0 such that H n ( x ) ∼ c log n as n → ∞ for almost all sequences x . Recently Devroye and Goudjil have shown that if the sequences x are Weyl sequences, i.e., defined by x k = { α k } , k = 1 , 2 , ⋯ , and α is distributed uniformly at random on [ 0 , 1 ] then H n ( x ) ∼ ( 12 / π 2 ) log n log log n as n → ∞ in probability. In this paper we consider the class of all uniformly distributed sequences x , and we show that the only behaviour that is excluded by the equidistribution of x is that of the worst case, i.e., for these x we necessarily have H n ( x ) = o ( n ) as n → ∞ .