implemented leadingWidth and automaticallyImplyLeading options (#136165)

Before:
![Screenshot 2023-10-09 at 01 32 33](https://github.com/flutter/flutter/assets/17102578/cb90423c-8d8d-40d5-afd0-5aab23b9930b)

After:

![Screenshot 2023-10-09 at 01 33 01](https://github.com/flutter/flutter/assets/17102578/3ba84243-42b8-4e02-b064-70618e21a305)

![Screenshot 2023-10-09 at 01 32 49](https://github.com/flutter/flutter/assets/17102578/4f311d74-f6a8-4f32-9974-109bd7f55812)

Fixed #136164

packages/flutter/lib/src/material/app_bar.dart

@@ -277,8 +277,8 @@ class AppBar extends StatefulWidget implements PreferredSizeWidget {
   /// {@template flutter.material.appbar.automaticallyImplyLeading}
   /// Controls whether we should try to imply the leading widget if null.
   ///
-  /// If true and [leading] is null, automatically try to deduce what the leading
+  /// If true and [AppBar.leading] is null, automatically try to deduce what the leading
-  /// widget should be. If false and [leading] is null, leading space is given to [title].
+  /// widget should be. If false and [AppBar.leading] is null, leading space is given to [AppBar.title].
   /// If leading widget is not null, this parameter has no effect.
   /// {@endtemplate}
   final bool automaticallyImplyLeading;
@@ -642,9 +642,9 @@ class AppBar extends StatefulWidget implements PreferredSizeWidget {
   final double? toolbarHeight;
 
   /// {@template flutter.material.appbar.leadingWidth}
-  /// Defines the width of [leading] widget.
+  /// Defines the width of [AppBar.leading] widget.
   ///
-  /// By default, the value of [leadingWidth] is 56.0.
+  /// By default, the value of [AppBar.leadingWidth] is 56.0.
   /// {@endtemplate}
   final double? leadingWidth;
 

packages/flutter/lib/src/material/search.dart

@@ -188,6 +188,12 @@ abstract class SearchDelegate<T> {
   ///  * [AppBar.leading], the intended use for the return value of this method.
   Widget? buildLeading(BuildContext context);
 
+  /// {@macro flutter.material.appbar.automaticallyImplyLeading}
+  bool? automaticallyImplyLeading;
+
+  /// {@macro flutter.material.appbar.leadingWidth}
+  double? leadingWidth;
+
   /// Widgets to display after the search query in the [AppBar].
   ///
   /// If the [query] is not empty, this should typically contain a button to
@@ -592,6 +598,8 @@ class _SearchPageState<T> extends State<_SearchPage<T>> {
         data: theme,
         child: Scaffold(
           appBar: AppBar(
+            leadingWidth: widget.delegate.leadingWidth,
+            automaticallyImplyLeading: widget.delegate.automaticallyImplyLeading ?? true,
             leading: widget.delegate.buildLeading(context),
             title: TextField(
               controller: widget.delegate._queryTextController,

packages/flutter/test/material/search_test.dart

@@ -1025,7 +1025,31 @@ void main() {
     expect(selectedResults, <String>['Result']);
   });
 
-  testWidgetsWithLeakTracking('showSearch with useRootNavigator', (WidgetTester tester) async {
+  testWidgets('Leading width size is 16', (WidgetTester tester) async {
+    final _TestSearchDelegate delegate = _TestSearchDelegate();
+    final List<String> selectedResults = <String>[];
+    delegate.leadingWidth = 16;
+
+    await tester.pumpWidget(TestHomePage(
+      delegate: delegate,
+      results: selectedResults,
+    ));
+
+    // Open the search page with check leading width smaller than 16.
+    await tester.tap(find.byTooltip('Search'));
+    await tester.pumpAndSettle();
+    await tester.tapAt(const Offset(16, 16));
+    expect(find.text('Suggestions'), findsOneWidget);
+    final Finder appBarFinder = find.byType(AppBar);
+    final AppBar appBar = tester.widget<AppBar>(appBarFinder);
+    expect(appBar.leadingWidth, 16);
+    await tester.tapAt(const Offset(8, 16));
+    await tester.pumpAndSettle();
+    expect(find.text('Suggestions'), findsNothing);
+    expect(find.text('HomeBody'), findsOneWidget);
+  });
+
+ testWidgetsWithLeakTracking('showSearch with useRootNavigator', (WidgetTester tester) async {
     final _MyNavigatorObserver rootObserver = _MyNavigatorObserver();
     final _MyNavigatorObserver localObserver = _MyNavigatorObserver();
 
@@ -1066,20 +1090,33 @@ void main() {
     expect(rootObserver.pushCount, 0);
     expect(localObserver.pushCount, 0);
 
-    // showSearch normal and back
+    // showSearch normal and back.
     await tester.tap(find.text('showSearchLocalNavigator'));
     await tester.pumpAndSettle();
+    final Finder backButtonFinder = find.byType(BackButton);
+    expect(backButtonFinder, findsWidgets);
     await tester.tap(find.byTooltip('Close'));
     await tester.pumpAndSettle();
     expect(rootObserver.pushCount, 0);
     expect(localObserver.pushCount, 1);
 
-    // showSearch with rootNavigator
+    // showSearch with rootNavigator.
     await tester.tap(find.text('showSearchRootNavigator'));
     await tester.pumpAndSettle();
     await tester.tap(find.byTooltip('Close'));
     await tester.pumpAndSettle();
-    expect(rootObserver.pushCount, 1);
+
+    // showSearch without back button.
+    delegate.automaticallyImplyLeading = false;
+    await tester.tap(find.text('showSearchRootNavigator'));
+    await tester.pumpAndSettle();
+    final Finder appBarFinder = find.byType(AppBar);
+    final AppBar appBar = tester.widget<AppBar>(appBarFinder);
+    expect(appBar.automaticallyImplyLeading, false);
+    expect(find.byTooltip('Back'), findsNothing);
+    await tester.tap(find.byTooltip('Close'));
+    await tester.pumpAndSettle();
+    expect(rootObserver.pushCount, 2);
     expect(localObserver.pushCount, 1);
   });