Add tree organization system with categories, tags, and folders

Features:
- Categories: Global and team-specific tree categorization (admin-managed)
- Tags: Flexible tree tagging with autocomplete (author + admin)
- User folders: Personal tree collections with subfolder support
  - Hierarchical structure (max 3 levels deep)
  - Right-click context menu for folder management
  - Cascade delete for subfolders
- Filter trees by category, tags, and folder in library view

Backend:
- New models: Category, Tag, UserFolder with relationships
- New API endpoints for categories, tags, and folders
- Tree organization migrations (005, 006)

Frontend:
- FolderSidebar with hierarchical folder tree
- FolderEditModal for create/edit with color picker
- AddToFolderMenu for quick tree organization
- TagInput with autocomplete and TagBadges display
- Updated TreeMetadataForm and TreeLibraryPage

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

backend/app/schemas/folder.py

@@ -0,0 +1,62 @@
+from datetime import datetime
+from typing import Optional
+from uuid import UUID
+from pydantic import BaseModel, Field
+import re
+
+
+# Valid hex color pattern
+HEX_COLOR_PATTERN = r'^#[0-9A-Fa-f]{6}$'
+
+
+class FolderBase(BaseModel):
+    name: str = Field(..., min_length=1, max_length=100)
+    color: str = Field("#6366f1", pattern=HEX_COLOR_PATTERN)
+    icon: str = Field("folder", max_length=50)
+
+
+class FolderCreate(FolderBase):
+    parent_id: Optional[UUID] = Field(None, description="Parent folder ID for creating subfolders")
+
+
+class FolderUpdate(BaseModel):
+    name: Optional[str] = Field(None, min_length=1, max_length=100)
+    color: Optional[str] = Field(None, pattern=HEX_COLOR_PATTERN)
+    icon: Optional[str] = Field(None, max_length=50)
+    display_order: Optional[int] = None
+    parent_id: Optional[UUID] = Field(None, description="Parent folder ID to move folder")
+
+
+class FolderResponse(FolderBase):
+    id: UUID
+    parent_id: Optional[UUID] = None
+    display_order: int
+    tree_count: int = 0  # Computed field
+    created_at: datetime
+    updated_at: datetime
+
+    class Config:
+        from_attributes = True
+
+
+class FolderListResponse(BaseModel):
+    id: UUID
+    name: str
+    color: str
+    icon: str
+    parent_id: Optional[UUID] = None
+    display_order: int
+    tree_count: int = 0
+
+    class Config:
+        from_attributes = True
+
+
+class FolderReorderRequest(BaseModel):
+    """Request body for reordering folders."""
+    folder_ids: list[UUID] = Field(..., min_length=1, description="Folder IDs in desired order")
+
+
+class FolderTreeRequest(BaseModel):
+    """Request body for adding a tree to a folder."""
+    tree_id: UUID