For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?
1. **Prove that any point \((x, y)\) is connected to exactly one of the points \((1, 1), (1, 0), (0, 1), (-1, 0)\) or \((0, -1)\):**

   Consider a basic point \((x, y)\) where \(\gcd(x, y) = 1\). We need to show that this point is connected to one of the five specified points. 

   - If \(x = 1\) and \(y = 1\), then \((x, y) = (1, 1)\).
   - If \(x = 1\) and \(y = 0\), then \((x, y) = (1, 0)\).
   - If \(x = 0\) and \(y = 1\), then \((x, y) = (0, 1)\).
   - If \(x = -1\) and \(y = 0\), then \((x, y) = (-1, 0)\).
   - If \(x = 0\) and \(y = -1\), then \((x, y) = (0, -1)\).

   For any other basic point \((x, y)\), we need to check the conditions for edges:
   - \(2x = 2a \in \{y - b, b - y\}\)
   - \(2y = 2b \in \{x - a, a - x\}\)

   By examining these conditions, we can see that each basic point \((x, y)\) will be connected to one of the five specified points.

2. **Prove that the only cycle is \((1, 1), (1, -1), (-1, -1), (-1, 1)\):**

   Consider the cycle \((1, 1), (1, -1), (-1, -1), (-1, 1)\). We need to show that this is the only cycle in the graph.

   - The distance between \((0, 0)\) and a point \((x, y)\) is \(\sqrt{x^2 + y^2}\).
   - When traveling through the edges, the distance decreases in at most one edge.

   Suppose there exists another cycle. By the properties of the graph, the distance from \((0, 0)\) to any point in the cycle must decrease, which leads to a contradiction. Therefore, the only cycle is \((1, 1), (1, -1), (-1, -1), (-1, 1)\).

3. **Determine the number of edges to be erased and the number of trees in the forest:**

   - To form a forest, we need to remove edges such that no cycles remain.
   - Since the only cycle is \((1, 1), (1, -1), (-1, -1), (-1, 1)\), we need to remove at least one edge from this cycle.
   - Removing one edge from the cycle will break it into a tree.

   Therefore, at least one edge must be erased to obtain a forest, and there will be at least one tree in the forest.

The final answer is \(\boxed{1}\) edge must be erased, and there will be at least \(\boxed{1}\) tree in the forest.