Charcoal, 82 bytes

ＷＳ⊞υι≔⟦⟦⟧⟧θＦＬυＦ⌕Ａ§υιT«≔⟦⟧ηＦθ«υＦλ«Ｊ§μ⁰§μ¹A»Ｆ⁴«ＪκιＭ✳⊗μ¿›⁼.ＫＫ№ＫＭA⊞η⁺λ⟦⟦ⅈⅉ⟧⟧»⎚»≔ηθ»ＩＬθ

Try it online! Link is to verbose version of code. Takes input as newline-terminated strings and outputs the number of solutions. Explanation:

ＷＳ⊞υι

Read in the grid.

≔⟦⟦⟧⟧θ

Start with 1 solution for 0 tents.

ＦＬυＦ⌕Ａ§υιT«

Loop over the positions of the trees.

≔⟦⟧η

No tent positions for this tree found so far.

Ｆθ«

Loop over the tent positions for the previous trees.

υ

Print the grid.

Ｆλ«Ｊ§μ⁰§μ¹A»

Print the tents for this partial solution.

Ｆ⁴«

Check the four orthogonal directions.

ＪκιＭ✳⊗μ

Move to the relevant adjacent square.

¿›⁼.ＫＫ№ＫＭA

If this square is empty and is not bordered by a tent, ...

⊞η⁺λ⟦⟦ⅈⅉ⟧⟧

... then append its position to the previous partial solution and add it to the list of new partial solution.

»⎚

Clear the canvas after testing this tree.

»≔ηθ

Save the new solutions as the current solutions.

»ＩＬθ

Print the final number of solutions.

Python 3.8 (pre-release), 253 244 bytes

from itertools import*
f=lambda b,h,w:all(set(t:=[i%w+i//w*1jfor i,e in enumerate(b)if e])&set(s:=[*map(sum,zip(t,T))])or~any(abs(a-b)<2for a,b in combinations(s,2))+all(h>a.imag>-1<a.real<w for a in s)for T in product(*[[1,1j,-1,-1j]]*sum(b)))

Try it online!

-6 bytes thanks to @user202729 (chain comparisons)

-3 bytes thanks to @ovs (1jfor; …or a+1^b → …or~a+b for "implies" boolean operator)

# Itertools for combinations and product
from itertools import*
f=lambda b,h,w: all(
    # Test if a given set of tent position deltas works:
    # Positions are complex numbers: real part increasing to the right, imaginary part increasing down
    # (De Morgan shortened, so many expressions negated)
        # No tree is on a tent:
            # t:=Tree positions (1s)
            set(t:=[i%w+i//w*1j for i,e in enumerate(b)if e])
            # s:=Tent positions as sum of tree positions and deltas
            & set(s:=[*map(sum,zip(t,T))])
        # and difference between all distinct pairs oftrees is at least 2:
            or any(abs(a-b)<2for a,b in combinations(s,2))
        # and all trees are within rectangular boundary
            # (Using Python 2's quirky complex floordiv doesn't work since those return complex nums,
            # which don't have a total order.
            # Plus Python 38 has saves so much here; using 2 would be a waste anyway)
            >= all(h > a.imag > -1 < a.real < w for a in s)
    # For each possible delta (four directions, distance 1)
    # sum(b) is the number of tents since each tent contributes 1
    for T in product(*[[1,1j,-1,-1j]]*sum(b))
)