a confusion

Consider a Poisson process of rate L on t>=0, N(1)=n. Let T1<...<Tn denote the times of these n points in [0,1]. Calculate the pdf of Tn.
Here is my answer:
F(t)=P(Tn<=t), {Tn<=t}={N((t,1])=0}, then I get answer. Pdf: f(y)= L*e^(-L+Ly), actually i just wanna ask, why it is not gamma distribution (n,L).. anyone can explain?? thanks.

however, if i take the time inversely, then Tn will be the first point, and the formula meets the gamma distribution... but, if i still take the time from 0 to 1, Tn is the last point, it doesnt meet the gamma. help appreciated..