fix(tree): address realtime-tree-server review findings

- make addTreeNode receivers idempotent (invalidateOnCreatePage guard +
  buildTree dedup) so the author's self-echo no longer duplicates the node
- broadcast realtime tree updates for bulk copy/duplicate and import via a
  root refetch: PAGE_CREATED now carries spaceId and the WS listener falls
  back to refetchRootTreeNodeEvent when no per-node snapshot is present
- remove the now-dead client-relay inbound path (isTreeEvent/handleTreeEvent)
  that remained a stale-restriction-cache attack surface
- honest string|null cast for a root move's parent id
- add tests: buildTree dedup; onPageCreated per-node vs refetch branching

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

apps/client/src/features/page/tree/utils/utils.test.ts

@@ -0,0 +1,40 @@
+import { describe, it, expect } from "vitest";
+import { buildTree } from "./utils";
+import type { IPage } from "@/features/page/types/page.types.ts";
+
+function page(id: string, position: string): IPage {
+  return {
+    id,
+    slugId: `slug-${id}`,
+    title: id.toUpperCase(),
+    icon: "",
+    position,
+    hasChildren: false,
+    spaceId: "space-1",
+    parentPageId: null as unknown as string,
+  } as IPage;
+}
+
+describe("buildTree", () => {
+  it("builds one node per unique page", () => {
+    const tree = buildTree([page("a", "a1"), page("b", "a2")]);
+    expect(tree.map((n) => n.id)).toEqual(["a", "b"]);
+  });
+
+  it("dedups a duplicate id so the tree has no duplicate node", () => {
+    // A realtime cache write could append a page twice; buildTree must not emit
+    // two references to the same node (which would crash the sidebar render with
+    // a duplicate React key).
+    const tree = buildTree([
+      page("a", "a1"),
+      page("b", "a2"),
+      page("a", "a1"), // duplicate id
+    ]);
+
+    expect(tree).toHaveLength(2);
+    expect(tree.map((n) => n.id).sort()).toEqual(["a", "b"]);
+    // No id appears more than once.
+    const ids = tree.map((n) => n.id);
+    expect(new Set(ids).size).toBe(ids.length);
+  });
+});