JavaScript (Node.js), 51 bytes

f=(n,s=0,l=1)=>n?s*f(n-=.5,s-1,l+1)+l*f(n,s+1,l):!s

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f=(
 n, // nodes: `2*n` is number of nodes remaining
 s=0, // slots: how many nodes with 1 child currently
 l=1 // leaves: how many leaf nodes currently
)=>
 n? // is all nodes placed on the tree?
 s*f(n-=.5,s-1,l+1)+ // chose a non-leaf node, place it as an child
 l*f(n,s+1,l): // chose a leaf node, place it as an child
 !s // make sure all non-leaf nodes have 2 children

JavaScript (Node.js), 51 bytes

f=(n,s=0,l=1)=>n?s*f(n-=.5,s-1,l+1)+l*f(n,s+1,l):!s

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This program output true for n=0. 1 more byte is required to convert it into 1.

f=(
 n, // nodes: `2*n` is number of nodes remaining
 s=0, // slots: how many nodes with 1 child currently
 l=1 // leaves: how many leaf nodes currently
)=>
 n? // is all nodes placed on the tree?
 s*f(n-=.5,s-1,l+1)+ // chose a non-leaf node, place it as an child
 l*f(n,s+1,l): // chose a leaf node, place it as an child
 !s // make sure all non-leaf nodes have 2 children

JavaScript (Node.js), 51 bytes

f=(n,s=0,l=1)=>n?s*f(n-=.5,s-1,l+1)+l*f(n,s+1,l):!s

Try it online!

JavaScript (Node.js), 51 bytes

f=(n,s=0,l=1)=>n?s*f(n-=.5,s-1,l+1)+l*f(n,s+1,l):!s

Try it online!

f=(
 n, // nodes: `2*n` is number of nodes remaining
 s=0, // slots: how many nodes with 1 child currently
 l=1 // leaves: how many leaf nodes currently
)=>
 n? // is all nodes placed on the tree?
 s*f(n-=.5,s-1,l+1)+ // chose a non-leaf node, place it as an child
 l*f(n,s+1,l): // chose a leaf node, place it as an child
 !s // make sure all non-leaf nodes have 2 children