How to solve the recurrence T(n)=T(n-1) + … T(1) +1?

Question

I need to find the complexity of an algorithm that involves the recurrence:

T(n) = T(n-1) + ... + T(1) + 1

T(n) is the time it takes to solve a problem of size n .

The master method doesn't apply here and I can't make a guess to use the substitution method (I don't want to use the substitution method anyway). I'm left with recursion tree method.

Since the number of children of each node isn't a constant, I'm finding it hard to keep track of how much each node contributes. What is the underlying pattern?

I understand that I have to find the number of nodes in a tree in which each node ( k ) has for its children all nodes numbered from 1 to k-1 .

Is it also possible to find the exact time T(n) given that formula?

Answer 1

Since T(n-1) = T(n-2) + ... + T(1) + 1

T(n) = T(n-1) + T(n-2) + ... + T(1) + 1
     = T(n-1) + T(n-1)
     = 2*T(n-1)

and T(1) = 1 => T(n) = 2^(n-1)