Correct : c

Explaination:

• Diameter of G = max_{u,v} d(u,v) = 30.
• Define the radius of G as rad(G) = min_{x} max_{y} d(x,y). It is always true that rad(G) ≤ diam(G) ≤ 2 · rad(G).
• From diam(G) = 30 we get rad(G) ≥ ⌈30/2⌉ = 15.
• For any root r, a BFS tree T rooted at r has height equal to the eccentricity of r, i.e. height(T) = max_{v} d(r,v) which is ≥ rad(G).
• Therefore for every BFS tree T we have height(T) ≥ rad(G) ≥ 15. Hence the height is at least 15.

Why other options are wrong (quick examples):

• A (exactly 15): Not guaranteed — if you root at an endpoint of a diameter pair, the BFS tree height can be 30.
• B (exactly 30): Not guaranteed — if you pick a center vertex (when rad(G)=15), a BFS tree can have height 15.
• D (at least 30): False — some BFS trees can have height 15, so "at least 30" is not true for every BFS tree.

Concrete illustrations:

• Take a simple path of 31 vertices (a path graph of length 30). Its diameter is 30. A BFS tree rooted at an endpoint has height 30; a BFS tree rooted at the middle vertex has height 15.