Finally, the first topographical map puzzle. There seems to be one or two of these every year, and this one seems a bit simpler than the others have been. A depth-first search of a relatively simple map; in this case, even fixed-length with a rule that, by defintion, can't go back on itself.

private int CountTrails(int x, int y)
{
    var trailEnds = new HashSet<(int, int)>();
    GetTrailEnds(x, y, 0, trailEnds);
    return trailEnds.Count;
}

private void GetTrailEnds(int x, int y, int height, HashSet<(int, int)> knownEnds)
{
    if (!Utils.IsInBounds(map, x, y) || map[x][y] != height) return;

    if (height == 9 && map[x][y] == height)
    {
        knownEnds.Add((x, y));
        return;
    }

    GetTrailEnds(x + 1, y, height + 1, knownEnds);
    GetTrailEnds(x - 1, y, height + 1, knownEnds);
    GetTrailEnds(x, y + 1, height + 1, knownEnds);
    GetTrailEnds(x, y - 1, height + 1, knownEnds);

}

Ha! I accidentally did part 2 initially because I misunderstood part 1. This is actually easier in a vaccum...

private int CountUniqueTrails(int x, int y, int height = 0)
{
    if (height == 9 && Utils.IsInBounds(map, x, y) && map[x][y] == height) return 1;
    if (!Utils.IsInBounds(map, x, y) || map[x][y] != height) return 0;

    return CountUniqueTrails(x + 1, y, height + 1)
            + CountUniqueTrails(x - 1, y, height + 1)
            + CountUniqueTrails(x, y + 1, height + 1)
            + CountUniqueTrails(x, y - 1, height + 1);
}

🎂