**Solution:**

Let's assume the height of the tree is h and the maximum height reached by the stone is H.

To solve this problem, we need to analyze the vertical and horizontal components of motion separately.

**Vertical Motion:**

1. The stone is thrown vertically upwards and reaches a maximum height of H.
2. The time taken to reach the maximum height can be calculated using the formula:

t = √(2H/g)

where g is the acceleration due to gravity.

This formula is derived from the kinematic equation:

H = (1/2)gt²

3. At the maximum height, the velocity of the stone becomes 0.

Using the equation:

v = u + gt

where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity, and t is the time taken,

we can calculate the initial vertical velocity of the stone (u) as:

0 = u + gt

u = -gt

4. The time taken for the stone to reach the bird after reaching the maximum height can be calculated using the same formula as step 2.

However, in this case, the initial height is H, and the final height is h.

So, the time taken (t') can be calculated as:

t' = √(2(H-h)/g)

5. The total time taken for the stone to hit the bird after being thrown can be calculated by adding the time taken to reach the maximum height (t) and the time taken to reach the bird from the maximum height (t').

Therefore, the total time taken (T) can be calculated as:

T = t + t'

**Horizontal Motion:**

1. The bird flies away horizontally with a constant velocity.
2. The horizontal distance traveled by the bird during the total time taken (T) can be calculated using the formula:

d = v * T

where d is the horizontal distance and v is the horizontal velocity of the bird.

**Calculating the Ratio of Horizontal Velocities:**

1. We need to find the ratio of the horizontal velocity of the stone to that of the bird.
2. The horizontal velocity of the stone can be calculated using the formula:

u_h = v_h

where u_h is the horizontal velocity of the stone and v_h is the horizontal velocity of the bird.

This is because the stone and the bird start at the same horizontal position and they both travel the same horizontal distance (d) in the same time (T).

3. Therefore, the ratio of the horizontal velocity of the stone to that of the bird is 1:1.

3/2