for any node in the tree, there are two possibilities

1. either the diameter is contained in one of the subtrees
2. or the node itself is at the top of the longest path in the tree that defines the diameter

in the second case, the diameter is equal to the sum of depth of the two deepest sub-trees of that node

the following algorithm calculates the diameter in $O(n)$ time

class Node:

def __init__(self, value, children):
self.value = value
self.children = children

@classmethod
def preorder(cls, lists):
return cls(
lists[0],
[cls.preorder(l) for l in lists[1:]]
)

def dd(root):
"""
returns depth, diameter for tree with given root node

>>> dd(Node.preorder([0, [1], [2]]))
(1, 2)
>>> dd(Node.preorder([1, [2, [3]], [4, [5]]]))
(2, 4)
>>> dd(Node.preorder([1, [2, [3, [4]], [5, [6, [7], [8]]]], [9]]))
(4, 5)
"""
d1 = d2 = -1
md = 0
for child in root.children:
depth, diameter = dd(child)
if diameter > md:
md = diameter
if depth >= d1:
d2 = d1
d1 = depth
elif depth > d2:
d2 = depth
return d1 + 1, max(md, d1 + d2 + 2)

if __name__ == "__main__":
import doctest
doctest.testmod()