Is there an upper bound of the number of induced subgraphs in a tree (i.e., connected acyclic undirected graph)?

For example, consider the following graph $G$ represented as edge list: $\{(S,T),(T,G),(T,B),(B,H),(H,Q)\}$. The number of connected subgraph (i.e., tree) is (represented by the set of vertices): $\{S\}$, $\{T\}$, $\{G\}$, $\{B\}$, $\{S,T\}$, $\{T,G\}$, $\{T,B\}$, $\{S,T,G\}$, $\{B,T,G\}$,$\{S,T,B\}$, $\{S,T,G,B\}$, $\{T,G,B,H\}$,$\{T,B,H,Q\}$, $\{S,T,B,H\}$,$\{S,T,B,H,G\}$,$\{T,G,B,H,Q\}$,$\{S,T,B,H,Q\}$, and $\{S,T,B,H,G,Q\}$.

I find two seemingly relevant answer that can directly answer my questions 1 and 2 but I'm not sure.

Is there an upper bound of the number of induced subgraphs in a tree (i.e., connected acyclic undirected graph)? The bound can be expressed in terms of vertices, edges, etc.

For example, consider the following graph $G$ represented as edge list: $\{(S,T),(T,G),(T,B),(B,H),(H,Q)\}$. The number of connected subgraph (i.e., tree) is (represented by the set of vertices): $\{S\}$, $\{T\}$, $\{G\}$, $\{B\}$, $\{S,T\}$, $\{T,G\}$, $\{T,B\}$, $\{S,T,G\}$, $\{B,T,G\}$,$\{S,T,B\}$, $\{S,T,G,B\}$, $\{T,G,B,H\}$,$\{T,B,H,Q\}$, $\{S,T,B,H\}$,$\{S,T,B,H,G\}$,$\{T,G,B,H,Q\}$,$\{S,T,B,H,Q\}$, and $\{S,T,B,H,G,Q\}$.

I find two seemingly relevant answer that can directly answer my questions 1 and 2 but I'm not sure.

I was wondering if there is any reference or results on the upper bound of the number of induced subgraphs in a connected acyclic graph (not directed) (i.e., tree). For example, consider the following graph $G$ represented as edge list: $\{(S,T),(T,G),(T,B),(B,H),(H,Q)\}$. The number of connected subgraph (i.e., tree) is (represented by the set of vertices): $\{S\}$, $\{T\}$, $\{G\}$, $\{B\}$, $\{S,T\}$, $\{T,G\}$, $\{T,B\}$, $\{S,T,G\}$, $\{B,T,G\}$,$\{S,T,B\}$, $\{S,T,G,B\}$, $\{T,G,B,H\}$,$\{T,B,H,Q\}$, $\{S,T,B,H\}$,$\{S,T,B,H,G\}$,$\{T,G,B,H,Q\}$,$\{S,T,B,H,Q\}$, and $\{S,T,B,H,G,Q\}$.

I find two seemingly relevant answer that can directly answer my questions 1 and 2 but I'm not sure.

Thanks!

Is there an upper bound of the number of induced subgraphs in a tree (i.e., connected acyclic undirected graph)?

For example, consider the following graph $G$ represented as edge list: $\{(S,T),(T,G),(T,B),(B,H),(H,Q)\}$. The number of connected subgraph (i.e., tree) is (represented by the set of vertices): $\{S\}$, $\{T\}$, $\{G\}$, $\{B\}$, $\{S,T\}$, $\{T,G\}$, $\{T,B\}$, $\{S,T,G\}$, $\{B,T,G\}$,$\{S,T,B\}$, $\{S,T,G,B\}$, $\{T,G,B,H\}$,$\{T,B,H,Q\}$, $\{S,T,B,H\}$,$\{S,T,B,H,G\}$,$\{T,G,B,H,Q\}$,$\{S,T,B,H,Q\}$, and $\{S,T,B,H,G,Q\}$.

I find two seemingly relevant answer that can directly answer my questions 1 and 2 but I'm not sure.

I was wondering if there is any reference or results on the upper bound of the number of connected subgraphs in a connected acyclic graph (not directed). For example, consider the following graph $G$ represented as edge list: $\{(S,T),(T,G),(T,B),(B,H),(H,Q)\}$. The number of connected subgraph (i.e., tree) is (represented by the set of vertices): $\{S\}$, $\{T\}$, $\{G\}$, $\{B\}$, $\{S,T\}$, $\{T,G\}$, $\{T,B\}$, $\{S,T,G\}$, $\{B,T,G\}$,$\{S,T,B\}$, $\{S,T,G,B\}$, $\{T,G,B,H\}$,$\{T,B,H,Q\}$, $\{S,T,B,H\}$,$\{S,T,B,H,G\}$,$\{T,G,B,H,Q\}$,$\{S,T,B,H,Q\}$, and $\{S,T,B,H,G,Q\}$.

I find two seemingly relevant answer that can directly answer my questions 1 and 2 but I'm not sure.

Thanks!

I was wondering if there is any reference or results on the upper bound of the number of induced subgraphs in a connected acyclic graph (not directed) (i.e., tree). For example, consider the following graph $G$ represented as edge list: $\{(S,T),(T,G),(T,B),(B,H),(H,Q)\}$. The number of connected subgraph (i.e., tree) is (represented by the set of vertices): $\{S\}$, $\{T\}$, $\{G\}$, $\{B\}$, $\{S,T\}$, $\{T,G\}$, $\{T,B\}$, $\{S,T,G\}$, $\{B,T,G\}$,$\{S,T,B\}$, $\{S,T,G,B\}$, $\{T,G,B,H\}$,$\{T,B,H,Q\}$, $\{S,T,B,H\}$,$\{S,T,B,H,G\}$,$\{T,G,B,H,Q\}$,$\{S,T,B,H,Q\}$, and $\{S,T,B,H,G,Q\}$.

I find two seemingly relevant answer that can directly answer my questions 1 and 2 but I'm not sure.

Thanks!