Add public/private visibility for trees

- Add is_public field to Tree model (private by default)
- Update access control: users see default trees, public trees, or their own
- Update all tree endpoints (list, search, get, categories) with new visibility logic
- Default/system trees are automatically marked as public
- Add migration 004 to add is_public column and update existing defaults
- Fix pydantic settings to ignore extra env vars (DATABASE_URL_SYNC)

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

backend/alembic/versions/004_add_tree_is_public.py

@@ -0,0 +1,32 @@
+"""Add is_public field to trees
+
+Revision ID: 004
+Revises: 003
+Create Date: 2026-02-01
+
+"""
+from typing import Sequence, Union
+
+from alembic import op
+import sqlalchemy as sa
+
+
+# revision identifiers, used by Alembic.
+revision: str = '004'
+down_revision: Union[str, None] = '003'
+branch_labels: Union[str, Sequence[str], None] = None
+depends_on: Union[str, Sequence[str], None] = None
+
+
+def upgrade() -> None:
+    # Add is_public column with default False (private by default)
+    op.add_column('trees', sa.Column('is_public', sa.Boolean(), nullable=False, server_default='false'))
+    op.create_index('ix_trees_is_public', 'trees', ['is_public'])
+
+    # Make existing default trees public
+    op.execute("UPDATE trees SET is_public = true WHERE is_default = true")
+
+
+def downgrade() -> None:
+    op.drop_index('ix_trees_is_public', table_name='trees')
+    op.drop_column('trees', 'is_public')