Looking more like her late father, Aussie actor Heath Ledger, every day, Matilda recently notched up two major milestones in any little girl's life.

The cute blonde celebrated her fourth birthday, and had her first day at preschool in Brooklyn. And she took it all in her happy-go-lucky stride.

Showing her resilience to the tragedy of losing her father 21 months ago, Matilda goofed around on a trip to the park in New York with her mum Michelle Williams last week.

Playing in a striped dress, knee-high socks and her favourite pair of fringed ankle boots, Matilda jumped, slid and climbed over every piece of available equipment.

And while doting mum Michelle, 29, was there with a helping hand to tackle the trickier obstacles, she seemed comfortable to let Matilda run free around the playground.

Matilda is understood to have spoken to her Perth-based grandfather Kim Ledger, whom she calls Papa, on her birthday, but will not see her Australian relatives until later this year, at Christmas.

"Unfortunately, we won't be seeing her [for her birthday], but we will be seeing her at Christmas," Heath's dad Kim told Perth's Sunday Times newspaper last week. "We'll be speaking to her for her birthday and all that sort of stuff…"