Question: Why is the recurrence relation of recursive factorial algorithm this?

```
T(n)=1 for n=0
T(n)=1+T(n-1) for n>0

```

Why is it not this?

```
T(n)=1 for n=0
T(n)=n*T(n-1) for n>0

```

Putting values of n i.e 1,2,3,4...... the second recurrence relation holds(The factorials are correctly calculated) not the first one.

Answer: 
Looks like T(n) is the recurrence relation of the *time complexity* of the recursive factorial algorithm, assuming constant time multiplication. Perhaps you misread your source?
I assume that you have bad information. The second recurrence relation you cite *is* the correct one, as you have observed. The first one just generates the natural numbers.