Complexity analysis of a recursion

A recursive algorithm has the following computation tree. The label on the node indicates the number of outgoing edges. This algorithm takes an input of length $$n$$ and at each processing, it results in $$n - 1$$ recursive calls. For each these calls/nodes, the size of the input decreases by $$1$$ from left to right. This pattern repeats.

I'm trying to figure out how to come up with a complexity analysis for this. It feels like there's some mathematical pattern/sequence here but I'm not able to figure it out.

Vertices in your tree correspond to decreasing sequences starting at $$n$$. We can identify each such subsequence with a subset of $$\{0,\ldots,n-1\}$$, and consequently, your tree contains $$2^n$$ vertices. Consequently, assuming the processing per node is $$O(1)$$, your algorithm runs in time $$\Theta(2^n)$$.
The same method also allows us to count directly the number of vertices labelled $$m$$: there are $$2^{n-m-1}$$ of these for $$m \in \{0,\ldots,n-1\}$$, and one labelled $$n$$ (the root).
Here is another way to solve this, using a recurrence. Let $$T(n)$$ be the size of the tree when the root is labelled $$n$$. Then $$T(0) = 1$$ and $$T(n + 1) = 1 + \sum_{m=0}^n T(m).$$ This formula actually works also for $$n = -1$$. In order to solve this recurrence, observe that $$T(n+1) = 1 + \sum_{m=0}^{n-1} T(m) + T(n) = 2T(n),$$ and so $$T(n) = 2^n T(0) = 2^n$$.