Remove vertices and triangles that are at a cirtain y value

Currently I'm generating a mesh at runtime based off of a heightmap. I have the UV unwraping and heightmap generation all working, but now I have an issue where I want there to be no mesh where there is black on the heightmap.

It's working OK so far, but there are some spots where the triangles come down to the 0/no data area, and I'm not sure how to fix it.

Here is my heightmap: (There are some "no data" values on the heightmap, it's square but the no data values show as transparent on here. In Unity they are converted to black)

When I generate my mesh, it comes out looking like this:

I don't want anything coming down to the "no data" area, but you can see there is stuff coming down. I believe this has to do with me generating all the vertices, and then if the area has data, adding the triangles, but I'm not sure how to fix it.

Here is my relevant function:

 void CreateMesh (Texture2D texture, int Startx, int Starty, int Width, int Height)
     {
         if (Width > 250 && Height > 250)
             return;
         List<Vector3> verts = new List<Vector3> ();
         List<int> tris = new List<int> ();
         List<Vector2> uvs = new List<Vector2> ();
         if (Startx + Width > texture.width)
             Width = texture.width - Startx;
         if (Starty + Height > texture.height)
             Height = texture.height - Starty;
         //uses 0x0 as the zero "no data" value
         float zero = texture.GetPixel (0, 0).grayscale * 100;
         //Bottom left section of the map, other sections are similar
         for (int i = 0; i < Width; i++) {
             for (int j = 0; j < Height; j++) {
                 float y = texture.GetPixel (Startx + i, Starty + j).grayscale * 100;
                 //Add each new vertex in the plane
                 verts.Add (new Vector3 (i, y, j));
                 uvs.Add (new Vector2 ((Startx + i) / ((float)texture.width), (Starty + j) / ((float)texture.height)));
 
                 //Skip if a new square on the plane hasn't been formed
                 if (i == 0 || j == 0)
                     continue;
                 if (texture.GetPixel(Startx+i,Starty+j) == Color.black || y < zero + ErrorMargin) continue;
 
                 //Adds the index of the three vertices in order to make up each of the two tris
                 tris.Add (Width * i + j); //Top right
                 tris.Add (Width * i + j - 1); //Bottom right
                 tris.Add (Width * (i - 1) + j - 1); //Bottom left - First triangle
                 tris.Add (Width * (i - 1) + j - 1); //Bottom left 
                 tris.Add (Width * (i - 1) + j); //Top left
                 tris.Add (Width * i + j); //Top right - Second triangle
             }
         }
         GameObject plane = new GameObject ("ProcPlane"); //Create GO and add necessary components
         plane.AddComponent<MeshFilter> ();
         MeshRenderer renderer = plane.AddComponent<MeshRenderer> () as MeshRenderer;
         renderer.material = mat;
         Mesh procMesh = new Mesh ();
         procMesh.vertices = verts.ToArray (); //Assign verts, uvs, and tris to the mesh
         procMesh.uv = uvs.ToArray ();
         procMesh.triangles = tris.ToArray ();
         procMesh.RecalculateNormals (); //Determines which way the triangles are facing
         plane.GetComponent<MeshFilter> ().mesh = procMesh; //Assign Mesh object to MeshFilter
         plane.transform.position = new Vector3 (Startx, 0, Starty);
         MeshCollider meshC = plane.AddComponent<MeshCollider> () as MeshCollider;
         meshC.sharedMesh = procMesh;
 
     }