How to get the most relevant category of a product on Amazon's Product Advertising API?

I am struggling to find out how to get a product's most relevant category path through Amazon's Product Advertising API. It seems that this information is not returned directly and needs to be determined by some client side logic?

Example:

http://ift.tt/1xHcctr

Category path displayed one the website:

Spielzeug › Sport & Outdoor › Tischbillard

Browse nodes returned from the api:

{
    "Ancestors": {
        "BrowseNode": {
            "Ancestors": {
                "BrowseNode": {
                    "Ancestors": {
                        "BrowseNode": {
                            "BrowseNodeId": "12950651",
                            "Name": "Spielzeug"
                        }
                    },
                    "BrowseNodeId": "12950661",
                    "IsCategoryRoot": "1",
                    "Name": "Kategorien"
                }
            },
            "BrowseNodeId": "360565031",
            "Name": "Sport & Outdoor"
        }
    },
    "BrowseNodeId": "2872727031",
    "Name": "Tischbillard"
},
{
    "Name": "Kategorien",
    "Ancestors": {
        "BrowseNode": {
            "BrowseNodeId": "16435051",
            "Name": "Sport & Freizeit"
        }
    },
    "Children": {
        "BrowseNode": [
            {
                "BrowseNodeId": "3772226031",
                "Name": "Sportbekleidung"
            },
            {
                "BrowseNodeId": "5143479031",
                "Name": "Sportschuhe"
            },
            {
                "BrowseNodeId": "190534011",
                "Name": "Sportelektronik"
            },
            {
                "BrowseNodeId": "190416011",
                "Name": "American Football"
            },
            {
                "BrowseNodeId": "16435131",
                "Name": "Angeln"
            },
            {
                "BrowseNodeId": "190423011",
                "Name": "Badminton"
            },
            {
                "BrowseNodeId": "190417011",
                "Name": "Baseball"
            },
            {
                "BrowseNodeId": "190418011",
                "Name": "Basketball"
            },
            {
                "BrowseNodeId": "190424011",
                "Name": "Billard"
            },
            {
                "BrowseNodeId": "248376011",
                "Name": "Bogenschießen & Waffensport"
            },
            {
                "BrowseNodeId": "248406011",
                "Name": "Bowling & Kegeln"
            },
            {
                "BrowseNodeId": "190425011",
                "Name": "Boxen"
            },
            {
                "BrowseNodeId": "16435151",
                "Name": "Camping & Outdoor"
            },
            {
                "BrowseNodeId": "451555031",
                "Name": "Cheerleading"
            },
            {
                "BrowseNodeId": "451582031",
                "Name": "Cricket"
            },
            {
                "BrowseNodeId": "190426011",
                "Name": "Darts"
            },
            {
                "BrowseNodeId": "189731031",
                "Name": "Event- & Reisegutscheine"
            },
            {
                "BrowseNodeId": "4256280031",
                "Name": "Fan-Shop"
            },
            {
                "BrowseNodeId": "451767031",
                "Name": "Fechten"
            },
            {
                "BrowseNodeId": "235141011",
                "Name": "Feldhockey"
            },
            {
                "BrowseNodeId": "16435171",
                "Name": "Fitness"
            },
            {
                "BrowseNodeId": "190432011",
                "Name": "Fun-Sportarten"
            },
            {
                "BrowseNodeId": "5143318031",
                "Name": "Futsal"
            },
            {
                "BrowseNodeId": "16435181",
                "Name": "Fußball"
            },
            {
                "BrowseNodeId": "16435191",
                "Name": "Golf"
            },
            {
                "BrowseNodeId": "190419011",
                "Name": "Handball"
            },
            {
                "BrowseNodeId": "451796031",
                "Name": "Inline-Skates & Rollschuhe"
            },
            {
                "BrowseNodeId": "248474011",
                "Name": "Jagd"
            },
            {
                "BrowseNodeId": "452697031",
                "Name": "Kampfsport"
            },
            {
                "BrowseNodeId": "451560031",
                "Name": "Klettern"
            },
            {
                "BrowseNodeId": "452680031",
                "Name": "Lacrosse"
            },
            {
                "BrowseNodeId": "190428011",
                "Name": "Leichtathletik"
            },
            {
                "BrowseNodeId": "5144499031",
                "Name": "Padel-Tennis"
            },
            {
                "BrowseNodeId": "451811031",
                "Name": "Paintball"
            },
            {
                "BrowseNodeId": "16435211",
                "Name": "Radsport"
            },
            {
                "BrowseNodeId": "16435221",
                "Name": "Reitsport"
            },
            {
                "BrowseNodeId": "452159031",
                "Name": "Rollhockey"
            },
            {
                "BrowseNodeId": "511836031",
                "Name": "Rucksäcke & Taschen"
            },
            {
                "BrowseNodeId": "452168031",
                "Name": "Rugby"
            },
            {
                "BrowseNodeId": "16435231",
                "Name": "Running"
            },
            {
                "BrowseNodeId": "452171031",
                "Name": "Scooter & Zubehör"
            },
            {
                "BrowseNodeId": "190608011",
                "Name": "Skateboarding"
            },
            {
                "BrowseNodeId": "460965031",
                "Name": "Snooker"
            },
            {
                "BrowseNodeId": "4543004031",
                "Name": "Sportbrillen"
            },
            {
                "BrowseNodeId": "190429011",
                "Name": "Squash"
            },
            {
                "BrowseNodeId": "451595031",
                "Name": "Tanzsport"
            },
            {
                "BrowseNodeId": "16435271",
                "Name": "Tennis"
            },
            {
                "BrowseNodeId": "190430011",
                "Name": "Tischtennis"
            },
            {
                "BrowseNodeId": "451774031",
                "Name": "Turnen"
            },
            {
                "BrowseNodeId": "190420011",
                "Name": "Volleyball"
            },
            {
                "BrowseNodeId": "3959757031",
                "Name": "Wassersport"
            },
            {
                "BrowseNodeId": "3024257031",
                "Name": "Wintersport"
            }
        ]
    },
    "IsCategoryRoot": "1",
    "BrowseNodeId": "16435121"
}

I can query the api and receive a collection/tree of browse nodes, which seems to contain the correct path (as [sometimes?] displayed on amazon's website).

But how do I determine the correct path?

I could not yet find any field in the API's response groups which would contain this information.

Can anyony give some hints?

Thx!