Question

Reading algorithms by self using Robert Sedwick book in C++

A recursive function that divides a problem of size N into two
independent (nonempty) parts that it solves recursively calls itself
less than N times.

If the parts are one of size k and one of size N-k, then the total
number of recursive calls that we use is T(n) = T(k) + T(n-k) + 1, for
N>=1 with T(1) = 0.

The solution T(N) = N-1 is immediate by induction. If the sizes sum to
a value less than N, the proof that the number of calls is less than
N-1 follows from same inductive argument.

My questions on above text are

1. How author came with solution T(N) = N-1 by induction? Please help me to understand.
2. What does author mean by “If the sizes sum to a value less than N, the proof that the number of calls is less than N-1 follows from same inductive argument” ?

I am new to mathematical induction so having difficulty in understanding.

Thanks for your time and help

Answer 1

(1) By induction:

T(1) = 0 (base)
T(N) = T(k) + T(N-k) + 1 (definition of problem)

We assume for each n < N, we get T(n) = n-1

Since both k and N-k are smaller then N, we get from the induction hypothesis:

T(N) = (k-1) + (N-k-1) + 1 = N-1
         ^        ^
        T(k)    T(N-k)

(2)
Using the same argument:
if

T(N) = T(k) + T(m) + 1 where k+m < N

Then the same proof will lead to T(N) < N-1