第一种解法

function findParentNode($nodes, $id)
{
    foreach ($nodes as $node) {
        if (in_array($id, $node['childrens'])) {
            return $node;
        }
    }
    return false;
}

function minimum2($nodes, $from, $to)
{
    $fromToTop = array($from);
    while ($parentNode = findParentNode($nodes, $from)) {
        $fromToTop[] = $parentNode['id'];
        $from = $parentNode['id'];
    }
    $toToTop = array($to);
    while ($parentNode = findParentNode($nodes, $to)) {
        $fromToTop[] = $parentNode['id'];
        $to = $parentNode['id'];
    }
    $path = false;
    $fromPath = array();
    $toPath = array();
    $countf = count($fromToTop);
    $countt = count($toToTop);
    for ($i=0; $i < $countf; $i++) {
        $fromPath[] = $fromToTop[$i];
        for ($j=0; $j < $countt; $j++) { 
            $toPath[] = $toToTop[$j];
            if ($fromToTop[$i] == $toToTop[$j]) {
                // 共同点。
                // 去掉一个共同点
                array_pop($fromPath);
                $path = implode('=>', $fromPath) . '=>' . implode('=>', $toPath);
                break 2;
            }
        }
        $toPath = array();
    }
    return $path;
}

第二种解法

function findParentNode($nodes, $id)
{
    foreach ($nodes as $node) {
        if (in_array($id, $node['childrens'])) {
            return $node;
        }
    }
    return false;
}

function minimum($nodes, $from, $to, $path = '', $tryParent = true)
{
    static $finds = array();
    static $excludeIds = array();
    if ($from == $to) {
        $path = $path . $from;
        $finds[] = $path;
        return $finds;
    }
    $path = $path . $from . '=>';

    $startNode = $nodes[$from];
    // 在子节点中找
    foreach ($startNode['childrens'] as $childId) {
        if (in_array($childId, $excludeIds)) {
            continue;
        }
        $childNode = $nodes[$childId];
        minimum($nodes, $childNode['id'], $to, $path, false);
    }

    $parentNode = findParentNode($nodes, $startNode['id']);
    $excludeIds[] = $startNode['id'];
    if ($tryParent && $parentNode) {
        minimum($nodes, $parentNode['id'], $to, $path, true);
    }

    usort($finds, function($a, $b) {
        $strLenA = strlen($a);
        $strLenB = strlen($b);
        if ($strLenA == $strLenB) {
            return 0;
        }
        return ($strLenA < $strLenB) ? -1 : 1;
    });
    return isset($finds[0]) ? $finds[0] : false;
}

测试

$nodes=array(
    array(
        'id' => 'A',
        'childrens' => array('B', 'C', 'D'),
    ),
    array(
        'id' => 'B',
        'childrens' => array('E', 'F', 'G'),
    ),
    array(
        'id' => 'C',
        'childrens' => array(),
    ),
    array(
        'id' => 'D',
        'childrens' => array('H', 'I', 'J'),
    ),
    array(
        'id' => 'E',
        'childrens' => array(),
    ),
    array(
        'id' => 'F',
        'childrens' => array('K', 'L', 'N'),
    ),
    array(
        'id' => 'G',
        'childrens' => array(),
    ),
    array(
        'id' => 'H',
        'childrens' => array('O', 'P', 'Q'),
    ),
    array(
        'id' => 'I',
        'childrens' => array(),
    ),
    array(
        'id' => 'J',
        'childrens' => array(),
    ),
    array(
        'id' => 'K',
        'childrens' => array(),
    ),
    array(
        'id' => 'L',
        'childrens' => array(),
    ),
    array(
        'id' => 'N',
        'childrens' => array('R', 'S', 'T'),
    ),
    array(
        'id' => 'O',
        'childrens' => array(),
    ),
    array(
        'id' => 'P',
        'childrens' => array(),
    ),
    array(
        'id' => 'Q',
        'childrens' => array(),
    ),
    array(
        'id' => 'R',
        'childrens' => array(),
    ),
    array(
        'id' => 'S',
        'childrens' => array(),
    ),
    array(
        'id' => 'T',
        'childrens' => array(),
    ),
);
$idNodeMap = array();
foreach($nodes as $key => $value) {
    $idNodeMap[$value['id']] = $value;
}

var_dump(minimum($idNodeMap, 'T', 'A'));
var_dump(minimum2($idNodeMap, 'T', 'A'));